Arsenal led comfortably early on thanks to goals from Sokratis and David Luiz, but Palace found a way back thanks to the ever-reliable Luka Milivojevic from the spot.
The hosts made the most of their early dominance by going in front, a corner from the right not cleared by Palace and when Granit Xhaka headed it on, Sokratis did well to angle a shot into the net in the 7th minute.
David Luiz doubled Arsenal lead two minutes after their first goal. Another corner from the Arsenal right was seemingly flicked on by Alexandre Lacazette and was swept in at the far post by former Chelsea defender.
Crystal Palace got some reprieve in the 32nd minute. Wilfred Zaha was brought down by Callum Chambers inside the Arsenal box. Referee Martin Atkinson initially booked Zaha for diving, but after a VAR check that decision was reversed and a penalty given.
Luka Milivojevic stepped up and slotted coolly past Bernd Leno to get Palace back in it.