Coming into the match on the back of four wins in their last five league games, including a 4-0 victory over Manchester United last time out, Everton had much the better of the action in the first half, but could not convert from their 11 shots.
After the break, the visitors continued to press with Dominic Calvert-Lewin firing into the side netting.
Everton went even closer in the 54th minute, but Bernard's powerful effort came out off the crossbar.
Palace goalkeeper Vicente Guaita then had to be at his best to deny substitute Cenk Tosun, while the hosts offered very little as an attacking force, seeing out the draw which means they have won just four times at home in the league this season.
The point helped Everton climb above Watford on goal difference into eighth position, but they are now four points behind Wolverhampton Wanderers in seventh, while Palace stay 12th.
(Reporting by Peter Hall; Editing by Christian Radnedge)
