Shkodran Mustafi opened the scoring for the Gunners, finishing well from a tight angle after Palace failed to clear their lines from a set-piece.
The Eagles equalised through Andros Townsend as the winger raced onto a cross from Wilfried Zaha to fire past Petr Cech.
Alexis Sanchez regained the lead for the Gunners, firing past Julian Speroni at the goalkeeper's near post.
The Chilean then sealed the result for Arsenal, latching onto Jack Wilshere's superb lofted pass to poke the ball past the goalkeeper before James Tomkins was able to nod home a consolation goal with minutes remaining.
Arsenal controlled possession
Early on it was established that Arsenal would try to dictate the pace of the game by dominating possession, while Palace looked sharp on the break down the left-hand side with Zaha and Ruben Loftus-Cheek.
Arsene Wenger returned to a three-at-the-back system as Calum Chambers was handed his first start in a Gunners shirt for 502 days.
The only real chance of the first 20 minutes though was created through some beautiful build-up play by the away side which ended up with Granit Xhaka’s shot from inside the box being well blocked.
Gunners find the breakthrough
However the opening goal came from a set-piece that Palace failed to clear, which ended up back at Alexandre Lacazette – his shot was palmed away by Speroni – but only to the feet of a grateful Mustafi who finished well at a tight angle.
Speroni will feel he could have done better and pushed the shot out of play, but manager Roy Hodgson will feel that his defenders should have cleared the ball before the ball got to Lacazette.
Palace have only kept one clean sheet in 13 home games since their 3-0 win against Arsenal in April, which will be a cause for concern for the Eagles’ manager.
Arsenal could have doubled their tally five minutes later, as Mesut Ozil raced onto a superb through ball by Sanchez however Speroni reacted well to divert the shot wide.
Zaha drilled a shot wide as Palace reminded their opponents that the home side are always going to be a threat on the counter, as Chambers picked up a booking early on which only further encouraged Zaha to target the former Southampton defender.
Palace bounce back
Arsenal came racing out of the blocks after the restart, as Hector Bellerin attempted to get on the end of a chipped ball from Sanchez but Loftus-Cheek did well to stop the Spaniard from a getting a shot away at close-range.
However it was Palace that equalised, as Zaha sized up Chambers before breezing past the defender and provided a low cross that Townsend ran onto and slammed the ball home from 12 yards.
The goal was Palace’s first shot on target all game, and was one of the few opportunities that Zaha was able to get his head up and run at a defender one-on-one.
Full of momentum, the hosts they had two opportunities to take the lead as Loftus-Cheek and then Jeffrey Schlupp had attempts from the left-hand channel that were blocked and sent wide respectively.
Arsenal lost their composure for the first 10 minutes of the second half, which Wenger will be displeased with after easing through the opening 45 minutes at Selhurst Park.
Gunners regain the lead
Fortunately for Arsenal they were able to regain their composure in time for Sanchez to power the ball home to give the Gunners the lead again.
The ball was teed up by Lacazette and went through Tomkins’ legs as Speroni couldn’t react in time to stop the ball going in at his near post.
The goalkeeper will be disappointed to be beaten at the near post, especially as the Argentine did get a hand on the effort but it wasn’t enough to push the ball wide.
Wilshere then turned provider for Sanchez with a quarterback-like pass to the winger from the half-way line before the Chilean brought the ball down well and stabbed the ball past Speroni.
Sanchez put in one of his better performances of the season tonight, not only scoring two vital goals but he also put in a shift defensively, often closing down Martin Kelly and Townsend before they could cross for Christian Benteke.
Zaha lost his marker at the back post and managed to get on the end of a cross by Bakary Sako but the winger could only send his header wide with 10 minutes remaining.
Palace rallied late on and got a late goal through Tomkins as the defender rose over Laurent Koscielny from a corner and gave Petr Cech no chance of saving his header before play fizzled out to full-time.
While Palace lie just one point above Bournemouth and the relegation zone. The Eagles have improved in recent months but are still looking over their shoulder and will be wary of the teams below them.