Jason Puncheon put the Eagles ahead on 23 minutes and Scott Dann doubled the advantage soon into the second half, heading home from Yannick Bolasie's corner.
Everton substitute Steven Naismith pulled one back but Cameron Jerome restored Palace's advantage and while Kevin Mirallas scored with four minutes remaining to set up a nervy finish, Palace held on to win and dent the Toffees' hopes of UEFA Champions League football.
Everton had been looking to make it eight league wins on the bounce and leapfrog Arsenal into fourth but instead they remain fifth - a point behind the Gunners, with both clubs having four more games to go.
Martinez told Sky Sports that conceding three goals showed his side were not at their best and he believed the situation had got to them.
"Defensively to concede three goals you'd expect the opposition to have more chances and more threat," he said, adding that conceding from a dead-ball situation from the corner was particularly disappointing.
"I did feel that the first half we had a little bit of a fearful look about our performance, we weren't ourselves at all.
"I think the reaction in the second half was terrific - it deserved something positive out of the game.
"And we created enough chances to do that, bearing in mind that Crystal Palace will always limit you to very little in front of goal because that's the way they are drilled and defensively they've got a terrific record.
"But I just feel that scoring two goals (to get back level) was the hardest part of this game.
"The disappointment is conceding three when they had four shots on target and that's down to probably our feeling of having something to lose today."