Dominic Calvert-Lewin’s fifth league goal of the season opened the scoring for the Toffees following neat build-up from James Rodriguez and Seamus Coleman.
Cheikhou Kouyate’s free header drew Palace level, but Richarlison restored the visitors’ lead from the penalty spot after Joel Ward was controversially adjudged to have handled Lucas Digne’s knockdown.
Everton had chances to extend their lead in the second half, but Calvert-Lewin shinned Rodriguez’s corner past the post before Colombian Rodriguez had a goal-bound shot blocked by Mamadou Sakho.
Palace saw more of the ball in the closing stages but failed to trouble Jordan Pickford in the Toffees’ goal.
The win lifts Carlo Ancelotti’s side to the top of the Premier League.
Fluid in attack, question marks in defence
This is the first time Everton have won their first five fixtures of a season since 1938 – and the first time they have started a top-flight campaign with three successive victories since 1993-94.
The Toffees were made to work hard against a Palace side who themselves had started the season with back-to-back league wins, but Ancelotti’s team fully deserved the three points.
The visitors – unchanged from last weekend’s win over West Brom – were quick to gain a stranglehold on possession and took the lead when Rodriguez’s first-time pass released Coleman, whose cut-back was tucked home confidently by Calvert-Lewin.
The 23-year-old is the first Everton player to score in three consecutive matches at the start of a season since Steven Naismith did so in 2014-15.
Calvert-Lewin should have doubled his tally in the second half when he beat two Eagles defenders to a Rodriguez corner, but he failed to direct his effort on target.
Michael Keane then appeared to get in the striker’s way from another Rodriguez delivery moments later, allowing Jordan Ayew to clear the danger.
Though pleased with his side’s overall performance, Ancelotti will be disappointed with the manner of Palace’s first-half equaliser, with Kouyate rising unmarked to nod Andros Townsend’s corner past Pickford.
Ward and Palace rue their luck
Ward, who was making his 250th appearance for the south London club, was visibly aggrieved at referee Kevin Friend’s decision to award Everton a 39th-minute penalty after consulting his pitch-side monitor – although the defender’s arm was away from his body when it made contact with Digne’s header.
The 30-year-old had survived an earlier video assistant referee check for a similar incident involving Richarlison, but Friend ruled in favour of the Palace man on that occasion.
There were positives for Roy Hodgson’s team, with summer signing Eberechi Eze producing a lively performance on his full Premier League debut for the club.
Other than Kouyate’s equaliser, however, the hosts failed to muster a shot on target on Pickford’s goal.
The makeshift centre-back went closest to a second Palace equaliser after half-time, but failed to hit the target with his far-post header.
The result stretches the Eagles’ winless home run against Everton to 10 league matches.