Both teams had chances to take maximum points at Selhurst Park, with Kagisho Dikgacoi and Jerome Thomas squandering opportunities for the hosts, while Phil Jagielka hit the crossbar for Everton.
Despite probably seeing the result as two points dropped, Everton have now gone five and a half hours without conceding and have lost just one of their opening 11 Premier League fixtures.
Keith Millen made one enforced change to the Palace side that lost 2-0 at West Brom last weekend, with Yannick Bolasie coming in for the injured Adlene Guedioura.
Roberto Martinez, meanwhile, decided to stick with the starting line-up that played out a goalless draw with Tottenham on Sunday.
After traffic around the stadium caused kick-off to be delayed, a defensive lapse in the fourth minute saw Marouane Chamakh charge down a Jagielka clearance, but Tim Howard rushed off his line to ensure the Moroccan could not capitalise.
Leon Osman then went close to breaking the deadlock for the visitors in the 12th minute as he got a boot to a Leighton Baines cut-back, guiding the ball narrowly wide of the target.
Dikgacoi had a glorious opportunity to put Palace ahead 10 minutes before the break, but the South African applied a dreadful header to fantastic Jerome Thomas cross from the left.
As Palace ramped up the pressure on Everton, Thomas forced Howard into an acrobatic save with a curling effort from 20 yards out two minutes later.
Osman tried his luck from distance on the stroke of half-time, but a Julian Speroni save ensured the teams remained gridlocked at the interval.
Thomas beat the offside trap to find himself through one-on-one with Howard eight minutes into the second half, but the winger sent his chip wide of the American's left post.
And the 30-year-old continued to look most likely to open the scoring as he was flagged for offside in the 67th minute after being sent clear on Howard once again.
Substitute Cameron Jerome created an angle for himself on the edge of the penalty area six minutes later, but skewed his effort well wide of Howard’s left-hand post.
Everton centre-back Jagielka had the crossbar shaking after 77 minutes as his diving header from a corner came agonisingly close to giving Martinez’s men the lead.
The visitors pushed hard for a winner in the closing stages, but were forced to settle for a draw as Palace collected just their fourth point of the season.
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