The New Testament makes the first connection itself. 1 Peter 3:20-21 says a few people, only eight souls, were saved through water, "And this water symbolizes the baptism that now saves you also." Peter then guards the sentence in the same breath: the saving is "not the removal of dirt from the body, but the pledge of a clear conscience toward God, through the resurrection of Jesus Christ." The water washes nobody. The resurrection does. Charles Spurgeon, preaching on Genesis 8:21 in 1865, gave the shape of it plainly: Noah was not saved by the world being gradually reformed, but by a sentence of condemnation, then death, burial and resurrection, going into the ark and coming out into a new world.
Genesis 6:16 puts one door in the side of the ark. Everything that lived went through it, and God closed it. Augustine, writing City of God in the early fifth century, read the ark as the church and the door as the wound in the side of the Crucified, "the door by which those who come to him enter in." That is Augustine reading the passage, not what Genesis says, and it is offered here as his reading. The plainer line needs no Father at all. Jesus says at John 10:9, "I am the door." Matthew Henry heard the same call in God's word to Noah at Genesis 7:1: "Christ is an ark already prepared, in whom alone we can be safe when death and judgment come."
Lamech named the boy for relief from ground the LORD had cursed, and the relief did not arrive. Genesis 8:21 has God lift the curse from the ground, and Genesis 9:20 has Noah go straight back to farming it and fall. Origen said the obvious thing in the third century: the words of Lamech are not appropriate to the ancient Noah, and "Jesus only has given rest to humanity and has freed the earth from the curse." Matthew 11:28 is where the name lands. This is a Christian reading of Genesis 5:29 held since the third century, and it is not a claim about what Lamech knew.
Jesus uses this man as the picture of His own coming. "As it was in the days of Noah, so will it be at the coming of the Son of Man" (Matthew 24:37). What He picks out is not the wickedness. It is that nobody noticed. They were eating and drinking, marrying and giving in marriage, ordinary life, right up to the day. Eusebius of Caesarea drew the structural point in the fourth century: the flood did not come before Noah entered the ark, and fire did not fall on Sodom before Lot left, so the righteous are separated before the judgment falls. J. C. Ryle added a caution worth keeping, that the passage is about the second coming and not about death, and that using these verses on a tombstone misreads them.
2 Peter 2:5 calls Noah a preacher of righteousness, and the Greek word is keryx, a herald. A herald announces a decision that has already been made rather than offering terms. It is the same root as the verb at 1 Peter 3:19. On that verse this site's posture receives first the reading that takes the spirits as the angels who sinned before the flood, held in chains as 2 Peter 2:4 and Jude 6 say, and the preaching as the risen Lord's proclamation of victory. Augustine read it instead as the preincarnate Christ preaching through Noah to men whose spirits are now in prison, and a third reading places the preaching in Hades between the death and the resurrection. Each stands with its holders, and on every main reading no second chance after death is taught.
Genesis 8:22 and 9:11 are the reason there is still a world for the gospel to be preached in. Seedtime and harvest do not cease and the flood does not come again. The promise retires the flood and not the judgment: 2 Peter 3:7 says the present heavens and earth are reserved for fire, kept for the day of judgment. Isaiah 54:9-10 is the Old Testament using the covenant the same way, and the logic is that God kept the first promise. Theodoret of Cyrus put it in one line in the fifth century: having the first covenant as a pledge, believe that you will always enjoy my love. Isaiah 54 is spoken to Israel, and where a teacher applies its comfort to all who believe, that is the teacher's application and not a transfer of the promise away from Israel.