Bidding to become only the third manager in Palace history to win their opening PL game in charge, Pierre Sage was nearly celebrating a perfect start inside three minutes, as talisman Jean-Philippe Mateta stabbed a glorious chance over from close range.
Keen to build on that early opportunity, the visitors threatened again soon after, with Eddie Nketiah seeing a low effort tipped onto the post by Jordan Pickford, while at the other end, Iliman Ndiaye was denied by a strong hand from Dean Henderson.
Both sides continued to probe for an opener as the first half progressed, and Mateta spurned another huge chance for the Eagles, firing agonisingly against the post with just Pickford to beat.
That miss proved to be costly, with Everton breaking the deadlock shortly before half-time, as Kiernan Dewsbury-Hall picked up a pass from Harrison Armstrong and arrowed a superb left-footed strike into the far corner beyond Henderson.
Having lost only one of their last 23 PL meetings with Palace, the Toffees came flying out of the blocks at the start of the second period, and their attacking intent was duly rewarded with a second goal in the 53rd minute.
Ndiaye was the architect, curling a sublime cross into the path of Thierno Barry, who made no mistake with a clinical header inside the six-yard box.
Armed with a two-goal cushion, David Moyes’ men remained on the front foot, as Ndiaye saw a volley thwarted by Henderson before Tyrique George missed the target with a poor effort across the face of goal.
The visitors pushed for an unlikely comeback in the closing stages, but Pickford remained alert to save Jørgen Strand Larsen’s header, as Moyes avoided going eight top-flight matches without a win in charge of the Toffees for the first time.
Flashscore Man of the Match: Kiernan Dewsbury-Hall (Everton)
