But if works are necessary for salvation, then isn't one saved by their works?
Not in the sense that St. Paul is talking about in Romans.
Rom 4:4 Now to the one who works, his wages are not counted as a gift but as his due.
Rom 4:5 And to the one who does not work but believes in him who justifies the ungodly, his faith is counted as righteousness,
Here St. Paul contrasts works and faith for salvation. The central point here and in the surrounding context is meriting or earning salvation -- expecting salvation as *payment* for the works. We do not believe that our works are earning salvation. Rather the work of God's grace in our hearts produces the righteousness in our that *is* salvation from sin.
Another way to talk about it is that Evangelical theology separates justification from sanctification and considers salvation to *be* justification, seen as a point event involving a judicial declaration but not necessarily a transformation. Orthodox thought has never separated the two. Salvation is justification plus sanctification as a continuous process because salvation involves not only being rescued from guilt but from sin itself.
Let's cut to the chase!
Jesus answered, “The work of God is this: to believe in the one he has sent.”
1 John 3:21-24
Dear friends, if our hearts do not condemn us, we have confidence before God and receive from him anything we ask, because we keep his commands and do what pleases him. And this is his command: to believe in the name of his Son, Jesus Christ, and to love one another as he commanded us. The one who keeps God’s commands lives in him, and he in them. And this is how we know that he lives in us: We know it by the Spirit he gave us.