I think what you express in your question is actually a form of the Nestorian heresy. You say that Jesus's "nature" died in our place and not his divine nature. The Nestorian controversy was named after the bishop Nestorius who taught that Mary gave birth to Christ and that Christ was crucified on the cross, and drew a distinction between the person of Christ and the divine nature in such a way as he could say that God's nature did not suffer on the cross and was not given birth to.
This teaching was condemned at the council of Ephesus and stands in opposition to the orthodox position that Jesus Christ is one person with both a divine and human nature and that those natures are united and unmixed in such a way so that Mary is called the Theotokos (birth-giver of God) and that we say God died on the cross for our sins.
This is expressed well in the recantation of Leporius recorded by St. John Cassian in Book 1 Chapter 5 of his work On the Incarnation of the Lord, Against Nestorius:
Therefore we confess that our Lord and God Jesus Christ the only Son of God, who for His own sake was begotten of the Father before all worlds, when in time He was for our sakes made man of the Holy Ghost and the ever-virgin Mary, was God at His birth; and while we confess the two substances of the flesh and the Word, we always acknowledge with pious belief and faith one and the same Person to be indivisibly God and man; and we say that from the time when He took upon Him flesh all the belonged to God was given to man, as all that belonged to man was joined to God. And in this sense 'the Word was made flesh:' not that He began by any conversion of change to be what He was not, but that by the Divine 'economy' the Word of the Father never left the Father, and yet vouchsafed to become truly man.
So yes certainly. Jesus Christ's divine nature has something to do with our atonement according to the canonical orthodox Christian faith. John Cassian offers many arguments in this work, but I will simply summarize the first one that directly pertains to the atonment. In Book 2 Chapter 4 of the same work St. John quotes Malachi 3:8 for an argument:
Shall a man pierce his God, for you are piercing me.
The interesting thing here is that most English translations appear to say "rob" and not "pierce". This may have something to do with the fact that the early Christians referred to the Septuagint and not the Hebrew text for the Old Testament; I'm not sure. But anyways, St. John says this is a prophecy of Christ's passion and that it clearly shows God contending with those persecuting him, not simply saying you have killed God's anointed, but that you have killed God himself.
As to how exactly the incarnation and crucifixion save us (it is not only the crucifixion, but the incarnation in its entirety that saves us), the best summary I can think of is this short sentence from St. Gregory of Nazianzus in his work To Cledonius the Priest Against Apollinarius. (http://www.ccel.org/ccel/schaff/npnf207.pdf)
For that which He has not assumed He has not healed; but that which is united to His Godhead is also saved.
So you see that it is because the divine nature enters all things pertaining to man (birth, life, work, eating, and death) that it can be transformed for our salvation. This is broader than just the atonement, but encompasses all of human life on earth. This is why the incarnation and nature of Christ were critically important doctrines to the Church and why bishops spent centuries arguing about the correct terminology and teachings about Christ, and sometimes were exiled, killed, or had their tongues cut out for teaching that Jesus Christ was fully man and fully God and truly suffered on the cross. Note that the Nicene creed is very careful to mention that God "suffered".