The home side had the better of the chances in the first half but was let down by poor finishing, though the visiting team could not make anything of it.
Both teams had chances to take maximum points at Selhurst Park, with Kagisho Dikgacoi and Jerome Thomas squandering opportunities for the home team, while Phil Jagielka hit the crossbar for Everton.
Despite probably seeing the result as two points dropped, Everton has now gone five and a half hours without conceding and has lost just one of its opening 11 Premier League fixtures.
Caretaker boss 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 lineup that played out a goalless draw with Tottenham on Sunday.
After traffic around the stadium caused kickoff to be delayed by 15 minutes, 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 capitalize.
Leon Osman then went close to breaking the deadlock for the visiting side 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 Jerome Thomas' fantastic 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 halftime, but a Julian Speroni save ensured the teams remained deadlocked 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.
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 post.
Everton center back Jagielka had the crossbar shaking after 77 minutes as his diving header from a corner came agonizingly close to giving Martinez's men the lead.
The visiting team pushed hard for a winner in the closing stages, but was forced to settle for a draw as Palace collected just its fourth point of the season.
Everton is fifth in the table, one point behind fourth-placed Chelsea, while Palace moved level on points with Sunderland but still sits bottom.