The game was delayed by 15 minutes due to a traffic incident near to the ground, but when proceedings did eventually get underway, it was Everton controlling possession with Kevin Mirallas the first to test Julian Speroni with a powerful low strike.
Palace looked to grow in confidence as the half progressed and they should have taken the lead when Kagisho Dikgacoi was found unmarked by Jerome Thomas at the back post, only for the midfielder to completely misjudge his header.
Thomas then forced Tim Howard into a fine acrobatic save with an effort from distance, before Marouane Chamakh headed wide from point-blank range as both sides went into the break level.
Thomas had Palace's best chance of the second half when he raced through on goal, but his chipped effort was miscued with Howard someway off his line.
Everton struggled to create too many chances of note after the interval, although Phil Jagielka did see his diving header rebound off the crossbar following Leighton Baines' cross.
The result ended a run of seven-straight Premier League defeats for the home side, but managerless Palace remain rock-bottom of the Premier League.
The club did appoint a new sporting director in the shape of recently-ousted Cardiff head of recruitment Iain Moody, and plan to unveil their new boss early next week despite maintaining a decision has not yet been made.
But whoever the new man is - Wales chief Chris Coleman and former Real Madrid assistant Aitor Karanka seem to be the frontrunners - they will have to somehow cure Palace's goalscoring problems.
Everton, by contrast, began the day with a chance to climb into the top four, but they have now drawn a blank in successive matches.
Romelu Lukaku started up front for the visitors despite injuring his knee in a collision with the head of Spurs goalkeeper Hugo Lloris last weekend.
But Lukaku never looked like adding to his five goals this season while this week it was the turn of Everton keeper Howard to take a whack on the head.
The American showed no after-effects with a stunning save to deny Thomas in the first half.
Predictably, Everton, looking for a third win in four matches, made all the early running with Leon Osman scuffing an early chance wide.
Then a slick move started by Mirallas, via Stephen Pienaar and Baines, ended with Osman slicing another effort the wrong side of the post.
Mirallas cut inside and sent in a low drive from the edge of the area which was palmed wide by Palace keeper Speroni, and Pienaar just failed to get on the end of Lukaku's fizzing cross.
But gradually it was Palace who took control of the game and they should have gone ahead when Dikgacoi met a cross from Thomas, only to make an almighty mess of his free header.
Howard was called into action after 38 minutes to pull off a superb diving save to prevent Thomas' curler nestling in the top corner.
And moments later Barry Bannan's cross was somehow headed wide from right in front of goal by Chamakh.
Worse was to come shortly after the break when Yannick Bolasie's through-ball caught the Everton defence out and Thomas raced clear one-on-one with Howard.
The winger's chip beat the advancing keeper, but unfortunately for Palace it also cleared the frame of the goal and another chance went begging.
Bolasie then curled a shot straight at Howard and substitute Cameron Jerome had an effort deflected wide.
Everton sensed they could snatch the victory and Jagielka's header from Baines' corner beat Speroni, but cannoned back off the crossbar.
Baines, Pienaar and Gerard Deulofeu peppered the home goal late on but the Palace defence did its job as the spoils were shared.