The hosts were dominant in the opening stages at Goodison Park and took a deserved lead through Romelu Lukaku in the ninth minute.
But that was as good as it got for Martinez's men, who saw the visitors hit back through Mile Jedinak's penalty, with Fraizer Campbell and Yannick Bolasie adding second-half strikes.
Leighton Baines did convert a late spot-kick to halve the deficit, but Palace stood firm as Everton fell to a second successive home defeat in the league.
"It was a really strange game," Martinez said. "We wanted to be sharp and we had to be bright on the ball because Crystal Palace are always really well defensively set up.
"I thought we were fantastic and the first 15 or 20 minutes were as good as we've played. We were controlling the game and looked fresh and scored a very good goal.
"I felt that at that point we lost a bit of intent and we didn't have that real drive about us to score that second goal, which was all we needed.
"Leon Osman had a good chance, if that goes in, it's game over. Then the penalty is an incredible reward for Crystal Palace. It's a long ball from Julian Speroni and we have to deal a lot better with it.
"That allowed Crystal Palace to have a lot of belief, and it forces a couple of errors which are very uncharacteristic for us.
"The character was incredible. We kept fighting and creating chances and probably did enough to get something from the game but it's very disappointing in the manner we started the game that we couldn't wrap up the performance with three points.
"Today just shows that this league is ruthless. It's a 10-month competition and every game is difficult. We were perfect for 15 minutes and we needed that two-goal cushion."
