Seamus Coleman’s 87th-minute goal left Sam Allardyce still awaiting his first win as Crystal Palace manager.
Everton thought they had taken the lead in the first half when Ross Barkley smashed home, but the Englishman was adjudged to have been offside when Romelu Lukaku's initial effort deflected off the post.
Palace hit the crossbar through a Christian Benteke header and were denied by a stunning save from Joel Robles as the match headed towards a 0-0 stalemate.
But three minutes from time Tom Davies found Coleman with a neat through-ball and the wing-back fired his fourth goal of the season to seal Everton's third PL win on the bounce.
The Eagles remain in the bottom three with Allardyce without a win in five as manager.
WHO'S GOING TO BE YOUR
MAN OF THE MATCH?
Vote at Full-time