The Toffees continued their unbeaten run against the Eagles that now stretches to nine games, but Marco Silva's side will be disappointed they failed to pick up all three points at Selhurst Park.
Everton started the game brightly and their high-pressing game made life difficult for the hosts in the early stages of the contest.
The visitors almost broke the deadlock just before the break when Richarlison whipped in a dangerous cross, only for the ball to go inches in front of the diving head of Dominic Calvert-Lewin.
Everton continued to press in the second-half but a resilient Palace remained solid at the back and managed to frustrate the Merseysiders into some ambitious efforts.
Substitute Cenk Tosun almost bagged the winner for the visitors in the 76th minute but his outrageous backheeled-effort produced a good save from Vicente Guaita.
The Eagles managed to see out the game, with the point earned seeing Roy Hodgsons side remain in 12th place in the Premier League table.