Regeneration isn't necessary for belief ... and comes AFTER.
Eph 1:13
In Him, you also, after listening to the message of truth, the gospel of your salvation—having also believed, you were sealed in Him with the Holy Spirit of promise,
14 who is given as a pledge of our inheritance, with a view to the redemption of God’s own possession, to the praise of His glory.
Notice it says "after" and "believed" is in the past tense? The scripture is clear - partial regeneration (regeneration of the spirit) comes AFTER faith, not before it. We are not fully regenerated (body soul & spirit) until the resurrection. Therefore, salvation is for the unregenerate.
See also John 20:31
31 but these are written that you may believe that Jesus is the Christ, the Son of God, and that believing you may have life in His name.