For more information related to this topic, check out Bob Yandian’s book
Understanding the End Times or his teaching series by the same title.

Tim, the passage to which you refer is from the Lord’s discourse (conversation styled teaching) from the Mount of Olives recorded in Matthew 24 & 25, Mark 13, and Luke 21.

Matthew 24:13:
       But he that shall endure unto the end, the same shall be saved.

The short answer to your question is “No.” This passage is not a reference to the Church. That does not mean that it might not have spiritual application to the believer’s life, but this admonition was directed to the Jewish people who would be entering into the Tribulation. Bob Yandian has written in detail about this passage in his book, Understanding the End Times. This specific passage is covered in Chapter 7, pages 103–117.

I will give you a brief summary of the message and context of Matthew 24:1–14, and some keys to understanding the overall message of these two chapters. Finally, I will address the issue of the security of the believer and our assurance of salvation through the completed work of Christ and the grace of God.

The Message of Matthew 24: 1–14
Having completed His discussions and debates with the religious leaders, Jesus left the temple to return to Bethany by way of the Mount of Olives (Matthew 24:3). The words Jesus had just spoken were still burning in His disciples' ears. He had denounced the nation and said it would be “desolate” (Matthew 23:38). If Jerusalem and the temple were destroyed, how would there be a nation for Messiah to rule? The disciples pointed out the buildings of the temple area to Jesus as if to impress Him with their magnificence. What could possibly happen to such impressive buildings, especially to the temple of God? Jesus' response troubled them: Not one stone here will be left on another; every one will be thrown down. The temple would be destroyed and Jerusalem with it. This, however, prompted the disciples to ask Jesus when all this would take place. As Jesus reached the Mount of Olives, He sat down and the disciples came to Him. Four disciples, Peter, James, John, and Andrew (Mark 13:3), plainly asked Jesus three questions:

1. When will this happen? That is, when will the temple be destroyed and not one stone left on another?
2. What will be the sign of Your coming…
3. …and of the end of the Age?

These questions prompted the following discussion by Jesus, commonly called the Olivet Discourse (Matthew 24:4-25:46).

Matthew 24:4-8: (Mark 13:5-8; Luke 21:8-11):

The questions the disciples asked of Jesus were related to the destruction of the temple and Jerusalem, the sign of the Lord's coming, and the end of the Age. They had nothing to do with the Church, which Jesus said He would build (Matthew 16:18). The Church is not present in chapters 24, and 25. The disciples' questions related to Jerusalem, Israel, and the Lord's Second Coming in glory to establish His kingdom. Actually, Matthew did not record Jesus' answer to the first question, but Luke did (Luke 21:20). The disciples felt that the destruction of Jerusalem, of which Jesus had spoken, would usher in the kingdom. They were thinking, no doubt, of Zechariah 14:1-2.

The destruction Jesus first referred to in Matthew 23:38 occurred in A.D. 67-70 (a destruction that is separate from the final one in Zechariah 14). This event was carried-out by the Romans and led to the dispersion of the Jewish nation, the destruction of the Temple, and the eventual building of an altar to Saturn on the TempleMount. Josephus records the incredible horrors of this time.
Jesus’ words to the disciples had a two-fold application (a common mark of prophetic teaching both in the Old and New Testaments). While the overall subject is a reference to the events preceding the Second Coming, there was immediate application to the lives of the first century Church. When they saw the signs of this Roman invasion, they were to take heed and flee the city. And, they did! The great majority of the Christians in Jerusalem fled the city before the worst of the destruction. The Jews who had rejected the Messiah, and His teachings also rejected the witness of the Church—and they died by the tens of thousands.

Jesus began to describe the events leading up to His return in glory and to indicate signs of that return. In this section (Matthew 24:4-8) He described the first half of the seven-year period preceding His second coming. That period is called the Seventieth Week of Daniel (Daniel 9:27), whichwill be characterized by:

1. False Christs (Matthew 24:4-5)
2. Wars and rumors of wars (Matthew 24:6) in which nations will rise against each other on a global scale (Matthew 24:7a)
3. Unusual disturbances in nature including famines and earthquakes (Matthew 24:7b)

These things (Matthew 24:8), Jesus said, will be the beginning of sorrows (birth pains). As a pregnant woman's birth pains indicate that her child will soon be born, so these universal conflicts and catastrophes will mean the end of the Church Age is near. This is the signal for the beginning of the Tribulation, called in the Old Testament the time of Jacob’s travail, or trouble. The above signs are not the wars, earthquakes, etc. that have occurred throughout time. These are supernatural in their origin—they are the beginnings of the “pouring-out of the wrath of God” upon the evil that has held the earth in bondage (see Romans 8:19–23).

The Church, as we know it, will not be present during this period. There will be believers—Jews and others who have accepted the salvation through believing in Jesus Christ as the Messiah—but the Church will have been removed in the event we call, the Rapture. This glorious event and the following period of “wrath” is spoken of in 1Thessalonians 4:17–5:11. Paul refers to this sudden removal of the Church from the earth by a Greek word translated “caught-up” in 4:17. The word means “to seize or take away” and is usually used in the sense of taking someone / something by force. Here Paul uses it in a gloriously positive manner. The believers who are alive at this time will be suddenly and with great power seized and taken away from the earth to be with the Lord—and no one can resist Him. This word was translated into Latin by Jerome and others who used the Latin word we translate as “rapture.”
As Paul continues into 1Thessalonians 5:1–11, he uses this as an encouragement to the believers. He is saying, “Don’t be discouraged, the ‘arrival of the Lord’ and the ‘catching-away’ of the Church has not happened yet—we are still here! When that happens the earth will be taken by surprise, but we shouldn’t be. We are to be alert, watching / waiting. To the unbelievers He will come as a ‘thief in the night’, but don’t be so distracted that His coming is a surprise to you. We are children of the light—we may not know the exact hour/day, but we should be aware that something is coming, and expect Him. This ‘removal’ will come before the wrath of God is poured-out upon the earth—for we the believers are not appointed to ‘wrath’, but to salvation.” (that’s the Geof Jackson paraphrase). Anyway, that is the message of this passage, and it is clear that this does not fit with the Church enduring through great times of tribulation which are meant to drive the Jews and others to acceptance of the Lord.

Matthew 24:9-14 (Mark 13:9-13; Luke 21:12-19):
Jesus began this next section (Matthew 24:9-14) with a time word, “Then.”At the middle point of the seven-year period preceding Jesus’ Second Coming, great distress will begin to be experienced by Israel. The Antichrist, who will have risen to power in the world and will have made a protective treaty with Israel, will break his agreement at that time (Daniel 9:27). He will bring great persecution on Israel (Daniel 7:25) and even establish his own center of worship in the temple in Jerusalem (2 Thessalonians 2:3-4). This will result in the death of many Jews (Matthew 24:9)and many people departing from the faith. Believing Jews will be betrayed by non-believers (v. 10), and many will be deceived by rising false prophets. Wickedness will increase, causing the love of most people (for Jehovah) to grow cold. Those who remain faithful to the Lord until the end of that period of time will be saved, that is, delivered through the tribulation and enter into the Millennium (Matthew 24:13).

This passage does not refer to a personal self-effort or endurance that results in one's eternal salvation, but to the physical deliverance of those who trust in the Savior during the Tribulation. Many of these will be persecuted / martyred, and great oppression will be directed at them. But a great multitude will endure through this seven-year period—they will enter the kingdom in physical bodies.

Also the gospel of the kingdom will be preached in the whole world during this period as a testimony to all nations. Though this will be a terrible time of persecution, the Lord will have servants who will witness and spread the good news concerning Christ and His soon-coming kingdom. This message will be similar to that preached by John the Baptist, Jesus, and the disciples at the beginning of Matthew's Gospel, but this message will clearly identify Jesus in His true character as the coming Messiah. This is not exactly the same message the church is proclaiming today. The message preached today in the Church Age and the message proclaimed in the Tribulation period calls for turning to the Savior for salvation. However, in the Tribulation the message will stress the coming kingdom, and those who then turn to the Savior for salvation will be allowed entrance into the kingdom. Apparently many will respond to that message (Revelation 7:9-10).

Matthew 24:15-26 (Mark 13:14-23; Luke 21:20-26):
Having given a brief overview of the entire Tribulation period prior to His return, Jesus then spoke of the greatest observable sign within that period, the abomination that causes desolation. This abomination was spoken of by Daniel (Daniel 9:27). It referred to the disruption of the Jewish worship which will be reinstituted in the Tribulation temple (Daniel 12:11) and the establishment of the worship of the world dictator, the Antichrist, in the temple. He will make the temple abominable (and therefore desolate) by setting up in the temple an image of himself to be worshiped (2 Thessalonians 2:4; Revelation 13:14-15). Such an event will be clearly recognizable by everyone. When that event occurs, “those…in Judea should flee to the mountains. They should not be concerned about taking anything with them or returning from the field for possessions, not even for a cloak.” The time following this event will be a time of great distress, unequaled from the beginning of the world and never to be equaled again (Jeremiah 30:7). The awful character of the Tribulation period cannot be truly grasped by anyone. This was why Jesus pointed out how difficult the time would be for pregnant women and nursing mothers (Matthew 24:19). He encouraged people to pray that their escape would not have to be in the winter when it would be difficult or on the Sabbath when travel would be limited (this is a clear indicator that Jesus was speaking directly to the Jewish people, not the Church—the Church will have been taken in the rapture before the beginning of these events). There is an encouraging note, however, for the Lord declared that those days would be cut short (v. 22). This means there will be a termination of this period of time. If it were to go on indefinitely, no one would survive. But the period will come to an end for the sake of the elect, those who will be redeemed / saved during the Tribulation and who will enter the kingdom. The elect of this Church Age will have already been raptured before the Tribulation. Much misinformation will be disseminated then for false Christs will be all around (vv. 23-24). They will all be preaching messages of salvation and performing signs and miracles, seeking to deceive even the elect. The Lord warned ahead of time not to be fooled for He would not be on earth working in that way.

The Security and Assurance of Our Salvation
While this passage in Matthew 24 and 25 is not specifically directed to the Church, we can learn from it. God, Himself, is in control of history, time, and the ultimate end of all things—man is NOT the Lord of the earth, and neither is Satan. What Jesus warned his disciples about actually happened within their generation—in part. And, the believers who listened were delivered from destruction: saved! Through the ages, the Church can draw great comfort from this and similar passages. God is in control, He is Faithful, and He will fulfill His promises! This message of encouragement and admonition is repeated in numerous passages throughout the New Testament:
       1 Thessalonians 4:17–5:11
       2 Thessalonians 1:6–2:17 
       Hebrews 10:23–25; 32–39
       2 Peter 3:1–15
Of course, there are many others scattered throughout the Letters to the Churches, and the teachings of the Lord in the Gospels, but these are wonderful promises that God is faithful to His word, and He will accomplish what He has begun.
This, now, brings me to my concluding point—our salvation is secure in His promise, not in our ability to “endure”. Consider:

Philippians 1:6: 
I am sure of this, that He who started a good work in you will carry it on to completion until the day of Christ Jesus.

Jude 1:24: 
Now to Him who is able to protect you from stumbling and to make you stand in the presence of His glory, blameless and with great joy, 25  to the only God our Savior, through Jesus Christ our Lord, be glory, majesty, power, and authority before all time, now, and forever. Amen.

John 10:27:
My sheep hear My voice, I know them, and they follow Me. 28  I give them eternal life, and they will never perish--ever! No one will snatch them out of My hand. 29  My Father, who has given them to Me, is greater than all. No one is able to snatch them out of the Father's hand.

John 6:37: 
Everyone the Father gives Me will come to Me, and the one who comes to Me I will never cast out. 38  For I have come down from heaven, not to do My will, but the will of Him who sent Me. 39  This is the will of Him who sent Me: that I should lose none of those He has given Me but should raise them up on the last day. 40  For this is the will of My Father: that everyone who sees the Son and believes in Him may have eternal life, and I will raise him up on the last day.

These passages, and many others, declare that our salvation is not dependent upon our ability to “stay saved,” but is a promise of His ability to “keep” us. This is why our salvation is such an awesome work of grace. There was nothing we could do to “get” saved, and there is nothing we can do to “keep” saved—He did it, and He gets all the Glory! Yes, certainly, God wants us to live-out our salvation, to produce works that are the result of faith, and to walk worthy (with equal weight) of the salvation we received. But, He will keep us!
I guess my favorite passages in regard to this wonderful promise are:

Hebrews 7:25: 
Therefore He is able also to save to the uttermost (completely, perfectly, finally, and for all time and eternity) those who come to God through Him, since He is always living to make petition to God and intercede with Him and intervene for them. (Amplified)

What an awesome declaration! He is ever “interceding for us” so that our salvation will be brought to its completion. I need to pray for issues that arise in my life, and for others who are facing difficulty and/or confusion. I need to pray for guidance and direction. But, I don’t need to pray that I can somehow endure to the end and obtain salvation. He HAS saved me, I accepted that by faith, I am eternally alive in Him, and He will bring me to His throne. And, besides, should there be something in my time in this life that would assault me and attempt to interfere with my salvation, His intercession is all-powerful so that nothing in this world WILL keep me from reaching His glorious goal.

Romans 8:34: 
Who is the one who condemns? Christ Jesus is the One who died, but even more, has been raised; He also is at the right hand of God and intercedes for us.  35  Who can separate us from the love of Christ?…

This great passage continues through verse 39 with the confident, bold, authoritative, victorious declaration that nothing in this life, past, present, future, above or below can keep us from obtaining the wonderful salvation that God has given us when we accepted Jesus Christ as Savior.

Geof W. Jackson, read bio
Director of the Grace School of Ministry/Director of Pastoral Care