JOEL - Yes. Actually the atonement of Jesus Christ encompasses His pre-mortal life; His ministry and life on the earth; his suffering for us in the garden and on the cross; His death and His resurection. Everything He did was directly related to the atonement.
The word "Atonement" is the act of unifying or bringing together what has been separated. The Atonement of Jesus Christ was necessary because of the separating transgression, or fall, of Adam, which brought both spiritual and physical death into the world. The atonement brings us back to God both spiritually and physically because of Christ's suffering, death, and resurrection.

Perhaps the question you mean to ask is at what moment did Christ actually suffer for our sins? Christ went first into the Garden of Gethsemane, there to bear the spiritual agony of soul only he could bear. He "began to be sorrowful and very heavy," saying to his three chief disciples, "My soul is exceeding sorrowful, unto death" (Mark 14:34). Leaving them to keep watch, he went further into the garden, where, according to the Book of Mormon, He would suffer "the pains of all men, yea, the pains of every living creature, both men, women, and children, who belong to the family of Adam" (2 Ne. 9:21). It was there in Gethsemane that his sweat was "as it were great drops of blood falling down to the ground" (Luke 22:44) for he bled "at every pore" (D&C 19:18). Christ then sealed His payment for our sins as He finally gave up His life on the cross.
Both His suffering in Gethsemane and death on the cross were necessary for the atonement to be complete, but the actual suffering for our sins was mainly experienced in the Garden. That was where the real price was paid for our sins because it was spiritual, voluntary, and directly related to payment for our sins according to eternal laws.
The suffering on the cross was mostly physical and was imposed by other men for their own reasons and was more related to His death, which was also an important part of the atonement.