The Gunners showed six changes to the team which won 1-0 at Watford on Monday evening, with Carl Jenkinson, Sead Kolasinac, Matteo Guendouzi, Mohamed Elneny, Mesut Ozil and Alex Lacazette recalled.
Joel Ward, Max Meyer and Cheikhou Kouyate were handed starts for the Eagles, with Jeffrey Schlupp out through injury and Andros Townsend and Patrick van Aanholt left on the bench.
Within the first minute Ozil tested Palace keeper Vicente Guaita with a low shot from 20 yards, but the game then slowed in pace with no clear openings at either end.
However, Elneny passed up a decent opportunity with 11 minutes played as he volleyed over from a Jenkinson cross, while at the other end Scott Dann headed down from a corner and forced Shkodran Mustafi into a timely clearance.
And 17 minutes in the visitors broke the deadlock, as a deep free-kick in from the right found an unmarked Christian Benteke, who headed home to break a goal drought stretching back to April 28th last year.
The Gunners took time to find any sort of response, with Konstantinos Mavropanos driving forward just past the half-hour mark, before dragging his shot well wide of target.
Just before the interval home keeper Bernd Leno made a superb double save to deny Kouyate and keep the deficit down to one goal.
Arsenal made a double change at the interval and two minutes into the second half they drew level, as Lacazette's fine pass released Ozil running into the left side of the area and he beat Guaita at his near post.
That prompted a period of Gunners dominance, with the visitors now pegged firmly back into their own half.
However, the Eagles broke forward to retake the lead just past the hour mark, with Benteke heading the ball on for Wilfried Zaha to break past the hesitant Mustafi and bury his shot past the advancing Leno.
And Palace added a third 20 minutes from time as Dann rose to meet a corner from the right and James McArthur got the final touch from close in to send the visiting fans into raptures.
Pierre-Emerick Aubameyang halved the deficit with just over ten minutes remaining as he rifled home an accurate, low right-footed finish from 12 yards out.
Palace held on through a nervy final few minutes, with Guaita making a couple of vital saves to ensure that his side are now definitely safe from the threat of relegation.