The Gunners were off to a quick start after only seven minutes when Sokratis Papastathopoulos punished poor defending by the visitors from a corner.
Lightning struck twice two minutes later from another Arsenal corner as David Luiz headed home to double the lead.
The video assistant referee decided Arsenal's Calum Chambers fouled Wilfried Zaha in the box after 32 minutes, allowing Luka Milivojevic to make it 2-1 from the penalty spot.
It was 2-2 seven minutes into the second half. Arsenal's defence appeared vulnerable and Jordan Ayew scored the equaliser.
The Gunners turned the screw late in the game and believed they had been rewarded when Sokratis appeared to have grabbed his brace with seven minutes to go, but VAR denied the goal for an earlier infringement.
Eagles manager Roy Hodgson would have been bitterly disappointed by his back four in the opening moments when the ball twice ended up in the back of their net from two corners.
The away side failed to deal with a pair of set pieces and were quickly 2-0 down after efforts from Sokratis and Luiz.
Both defenders converted simple opportunities, and Palace's defence were caught on their heels in situations they should have dealt with.
However, the Eagles fought back as the first half unfolded, and VAR overturned the referee's call in order to award a penalty.
Zaha was initially booked for simulation after a challenge by Chambers, but the caution was reversed when VAR intervened.
Milivojevic has consistently proved himself to be an excellent penalty taker for Hodgson's team, and the skipper made no mistake as he made it 2-1.
The goal motivated the visitors as the hosts regressed, and Palace came out for the second half with the wind in their sails.
An equaliser was the reward shortly after the restart, and the Gunners crumbled as Ayew brought the game back to 2-2.
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Arsenal went hunting a winner in the final stages of the game, and Palace goalkeeper Wayne Hennessey was forced to deny Alexandre Lacazette.
There was controversy late in the contest when Unai Emery's men thought they had won the points.
Further bad defending by the visitors saw Sokratis hit the back of the net for a second time, but VAR concluded the goal would not stand due to a foul in the build-up.
Palace will be delighted with a point after their early lapse, but it's back to the drawing board for Emery, who watched his side fail to convince in the Premier League once again.