Mikel Arteta's second-half penalty and a late goal from Olivier Giroud ensured Arsenal opened up a five-point gap at the top of the Premier League table.
Arteta, who saw red for a professional foul on former Arsenal teammate Marouane Chamakh, scored the opener before Giroud sealed the win late on.
The Gunners were restricted to just a handful of chances against Palace, as the Eagles put in a fighting display against Arsene Wenger's side.
Palace put in a far better performance than they did against Fulham, with the Eagles pressing Arsenal at every opportunity.
The home side's best chance came after 30 minutes when Barry Bannan's curling effort sailed just wide of Wojciech Szczesny's right-hand post.
Arsenal were awarded a penalty shortly after half-time when Serge Gnabry tricked his way into Palace's penalty area before having his legs taken out by Adlene Guedioura.
Gunners captain Arteta converted from 12 yards after referee Chris Foy was left with no choice but to award the Premier League leaders a penalty.
Arsenal went down to 10 men in the 65th minute when Arteta deliberately clipped the heels of Chamakh, who was running clear through on goal.
Palace bombarded Arsenal's goal after Arteta's dismissal - with Mile Jedinak and Joel Ward both going close to grabbing an equaliser.
However Giroud headed home from close range to seal his side the three points with three minutes to go.