This is true. Salvation involves the forgiveness of sins. But there's also a way Christmas saves us as well.
According to the church fathers, the Incarnation saves us ontologically.
Definition note: "ontology" is the branch of metaphysics dealing with the nature of being. So when I say Christmas saves us ontologically I'm saying that God does something to human being, to the nature and mode of our existence. Crudely, God saves "what we are made of." Specifically, after the Fall, humanity was in a weakened ontological state. We were separated from God, and therefore vulnerable to the forces of death, decay, and dissolution. As Athanasius describes the situation in On the Incarnation, "For the nature of created things, having come into being from nothing, is unstable, and is weak and mortal when considered by itself."
Consequently, separated from God humanity was dissolving, fading away. As Athanasius says, "For these reasons, then, with death holding greater sway and corruption remaining fast against human beings, the race of humans was perishing, and the human being, made rational and in the image, was disappearing, and the work made by God was being obliterated."
So, our predicament here was ontological. Constitutionally, human being was unstable, weak, mortal. We were fading, disappearing, on the road to oblivion.
But on Christmas, in the Incarnation, God reunites Himself, through the Son, with human being. When the Word is made flesh an ontological stabilization occurs, anchoring human being and saving us from dissolution and oblivion. As Athanasius says,
So seeing that all created nature according to its own definition is in a state of flux and dissolution, therefore to prevent this happening and the universe dissolving back into nothing, after making everything by his own eternal Word and bringing creation into existence, [God] did not abandon it to be carried away and suffer through its own nature, lest it run the risk of returning to nothing...lest it suffer what would happen...a relapse into non-existence, if it were not protected by the Word.And that's how Christmas saves us. Through the Incarnation, human being is reunited with God and is now stabilized and protected by the Word.