The atonement is linked to its application, but we must be careful not to conflate the extent with the application. As Shedd correctly states,
The expiation of sin is distinguishable from the pardon of it. The former, conceivably, might take place and the latter not. When Christ died on Calvary, the whole mass, so to speak, of human sin was expiated merely by that death; but the whole mass was not pardoned merely by that death. The claims of law and justice for the sins of the whole world were satisfied by the “offering of the body of Jesus Christ once for all” (Heb. 10:10); but the sins of every individual man were not forgiven and “blotted out” by this transaction. Still another transaction was requisite in order to this, namely, the work of the Holy Spirit in the heart of the sinner working faith in this expiatory offering and the declarative act of God saying “your sin is forgiven you.” The Son of God, after he had offered one sacrifice for sins forever, “sat down on the right hand of God” (10:12); but if the redeeming work of the Trinity had stopped at this point, not a soul of mankind would have been pardoned and justified, yet the expiatory value of the “one sacrifice” would have been just the same. (W. G. T. Shedd, Dogmatic Theology, 3:418).
Nowhere in Scripture are we told that atonement is equal to salvation. The benefits of the atonement must be applied to the individual to be efficacious, and such application is clearly conditioned in the NT upon faith in Christ. The cross itself, unapplied, saves no one. Salvation is both an objective and subjective reality. Salvation is effected not only through the death of Christ on the cross but also through the application of the benefits of His death by the Holy Spirit. We participate in the life of the triune God “through the work of Christ as the ground of its possibility, and through the agency of the Holy Spirit as its actualization.” (A. Thiselton, The Hermeneutics of Doctrine, 339, emphasis original.)