With neither side in good form, the cagey start was somewhat expected but the game almost came to life after 10 minutes. Jean-Philippe Mateta fizzed a pass into Odsonne Edouard, who found himself in space at the edge of the area, only for his daisy-cutter to be easily stopped by Jordan Pickford.
Everton were making a concerted effort to hit Dominic Calvert-Lewin early and often in an attempt to find a way through, and that breakthrough almost arrived when Dwight McNeil picked him out with a perfectly weighted cross but his powerful header lacked accuracy.
Another chance came and went for the Eagles after a swift passing move ended with Daniel Munoz laying the ball off for Jefferson Lerma, but the midfielder lost his composure and his first-time effort drifted harmlessly wide.
The break came at a good time for the Toffees as Palace were in the ascendancy, and half-time appeared to stall their momentum as the second half started as scrappily as the first.
Despite struggling from open play, the hosts looked dangerous from set pieces and should’ve led just after the hour mark.
Abdoulaye Doucoure was all alone at the back post and just had to divert the ball into the empty net, yet his finish lacked conviction and allowed Sam Johnstone to claw the ball to safety.
Sean Dyche’s side were made to pay almost immediately when Jordan Ayew produced a moment of real quality, picking up a loose ball on the edge of the area and unleashing a rocket into the far corner that left Pickford with no chance.
Just when it looked like the game was slipping away from the hosts, McNeil’s floated corner was missed by Johnstone, allowing Amadou Onana to nod the ball home from close range.
Despite now being winless in six games, Everton climb out of the relegation zone thanks to a superior goal difference over Luton Town. Meanwhile, Palace rise to 15th but fail to extend their cushion over the relegation places which remains at five points.
Flashscore Man of the Match: James Tarkowski (Everton)