Atonement means that humanity is completely at one with Jesus in the totality of his life, and enjoys all the benefits of salvation from this identification.
Agreement on this in the early Fathers of the Church is overwhelming:

1. St Gregory of Nazianzen: The Word in becoming incarnate “bears all me and mine in himself, that in himself he may exhaust the bad, as fire does wax, or as the sun does the mists of the earth; and that I may partake of his nature by the blending.” Or again: “So he is called man…that by himself he may sanctify humanity and be as it were a leaven to the whole lump; and that by uniting to himself what was condemned he may release it from condemnation, becoming for all men all that we are except sin – body, soul, mind, and in every part that death reaches.”

2. St Gregory of Nyssa: “Although Christ took our filth upon himself, nevertheless he is not himself defiled by the pollution, but in his own self he cleanses the filth, for it says, the light shone in the darkness, but the darkness did not overpower it.” And again: “Although he was made sin and a curse because of us, and took our weaknesses upon himself, yet he did not leave the sin and the curse and the weakness enveloping him unhealed…Whatever is weak in our nature and subject to death was united with his Deity and became what the Deity is.”

3. St Cyril of Alexandria: “There was no other way for the flesh to become life-giving, since by its own nature it is subject to the necessity of corruption, except that it became the very flesh of the Word who gives life to all things…There is nothing to surprise here. Just as fire has converse with materials that are not hot, yet renders them hot by abundantly introducing into them the inherent energy of its own power; then surely in an even greater degree, the Word who is God can introduce the life-giving power and energy of his own nature into the flesh of his own people.”

Atonement thus occurs for the Fathers through the dynamic of the incarnation itself, not by way of some extrinsic theory, i.e., satisfaction, penal substitution, and so on. Why, one wonders, did theology subsequently fail to reflect this?  I am not sure. Part of the reason, I suspect, lies in how the incarnation came to be largely understood.

With focus on the miracle of God becoming flesh in the birth of Jesus, the saving significance of the rest of Jesus’ life was overshadowed. With focus returned, so to speak, on the Cross, the climactic end of Jesus’ life, the impression de facto was that the real meaning of God’s identification lay at the beginning and at the end, not in the entire range of Jesus’ life.

Something else is underlined in the theology of Fathers, which is most important to note. It concerns the kind of humanity assumed by Jesus. This is not an ideal humanity, of the kind that preexisted the Fall.

The ancient maxim has it just right: “What has not been assumed has not been saved.” The humanity assumed is ours, i.e., humanity suffering from fear, distress, conflict, anxiety in the face of death, and separation from God – all the qualities of life affected by sin. God assumed it to reverse and transform it.

The picture of God thus disclosed is that of a God assuming all that flesh is heir to in the fullest possible solidarity with us. As St Gregory of Nazianzen put it: In becoming flesh, God “bears all me and mine in himself, that in himself he may exhaust the bad, as fire does wax, or as the sun does the mists of the earth; and that I may partake of his nature by the blending.”

Or again: “So he is called man…that by himself he may sanctify humanity and be as it were a leaven to the whole lump; and that by uniting to himself what was condemned he may release it from condemnation, becoming for all men all that we are except sin – body, soul, mind, and in every part that death reaches.”

This process described in Gregory can also be taken as a process of expiation. Expiation is a process separate or distinct from atonement. It is rather atonement viewed as something Jesus actively accomplishes, something he does, not something that simply happens to him.

What we note immediately in this connection is that Gregory never mentions or alludes to blood, though that is the main element the Catholic imagination has traditionally dwelt on when expiation came to mind. It’s why the Mass was referred to as an unbloody sacrifice. Blood was the essential element in expiatory sacrifice. Since the Mass reenacted the sacrifice, the representation could only be unbloody.

Gregory speaks instead of a process of exhaustion, and of uniting to oneself, in other words, a process of incorporation or absorption, which ends in a release from condemnation” The dynamic, in other words, remains within the context of identification. God in Jesus totally absorbs our sin, and thus takes it away, i.e. his absorption frees us from our sin and all its effects.

An excellent Biblical example of this process (and of how we too can expiate the sins of others) is given in the story of Joseph in Genesis 37. Joseph absorbs the sin of his brothers by exhausting it in his own person. His act (of which they are completely unaware) releases and delivers them from the condemnation their sin brought on themselves.

To conclude then: the Eucharist is the sacrament of Jesus’ sacrifice, i.e., of his total self-giving from birth to death. His sacrifice saves us, because in virtue of the incarnation we are transformed through our complete identification with him, and made, as St Paul expressed it, co-heirs with him of the divine life. We celebrate this sacrament in the ritual form of a meal.