Is the salvation that Christ gained for us on the cross given to us through the sacraments?
Yes, I think that’s fair to say. Although to be unambiguous about phraseology, some clarification is a good idea…
Here’s some relevant paragraphs from the CCC:
1076 The Church was made manifest to the world on the day of Pentecost by the outpouring of the Holy Spirit. The gift of the Spirit ushers in a new era in the “dispensation of the mystery” the age of the Church, during which Christ manifests, makes present, and communicates his work of salvation through the liturgy of his Church, "until he comes."In this age of the Church Christ now lives and acts in and with his Church, in a new way appropriate to this new age. He acts through the sacraments in what the common Tradition of the East and the West calls “the sacramental economy”; this is the communication (or “dispensation”) of the fruits of Christ’s Paschal mystery in the celebration of the Church’s “sacramental” liturgy.
1115 Jesus’ words and actions during his hidden life and public ministry were already salvific, for they anticipated the power of his Paschal mystery. They announced and prepared what he was going to give the Church when all was accomplished. The mysteries of Christ’s life are the foundations of what he would henceforth dispense in the sacraments, through the ministers of his Church, for “what was visible in our Savior has passed over into his mysteries.”
1210 Christ instituted the sacraments of the new law. There are seven: Baptism, Confirmation (or Chrismation), the Eucharist, Penance, the Anointing of the Sick, Holy Orders and Matrimony. The seven sacraments touch all the stages and all the important moments of Christian life: they give birth and increase, healing and mission to the Christian’s life of faith. There is thus a certain resemblance between the stages of natural life and the stages of the spiritual life.