In LotR, good did indeed destroy evil (or rather its boss) by destroying the one ring.
The only way the Ring was destroyed was by accident when Gollum attacked Frodo to claim the Ring. The Evil that the ring stoked in the hearts of those it touched is what ended up destroying it in the end, not the Good people who took it to Mount Doom.
> Tolkien in his letters insists that Frodo did not fail – that his captivity to the Ring was inevitable
> No theme may be played that hath not its uttermost source in me, nor can any alter the music in my despite. For he that attempteth this shall prove but mine instrument in the devising of things more wonderful, which he himself hath not imagined.
The idea being that the evil that Melkor commits ultimately just builds towards the creators greater and secret purpose. It's kind of an attempt to deal with the "problem of evil".
(I’m not a Christian though so somebody please correct me if I am wrong)