Mikel Arteta scored from the penalty spot before being sent off for bringing down Marouane Chamakh, but the Pole made some impressive saves before Olivier Giroud sealed the win.
The Spaniard stroked in a 47th-minute penalty to give the Gunners the lead after Serge Gnabry was fouled by Adlene Guedioura, but was then dismissed for bringing down Marouane Chamakh as the last man.
Wojciech Szczesny made several superb saves to prevent Palace finding an equalizer, and Olivier Giroud headed in Aaron Ramsey's scooped cross to seal the win late on.
Keith Millen, taking temporary charge of Palace following the departure of Ian Holloway, made six changes to the side beaten by Fulham, with Jerome Thomas, Barry Bannan and Chamakh coming into the side, while Mathieu Flamini and Santi Cazorla returned to Arsenal’s lineup in place of Jack Wilshere and Tomas Rosicky.
Arsenal created its first chance after just three minutes when Giroud headed Bacary Sagna's right-wing cross just over the crossbar.
The Gunners were forced to change things inside the first 10 minutes, though, Flamini replaced by Gnabry after he limped off with an apparent groin injury.
Palace competed well for large periods of the first half, and had opportunities to take the lead with Bannan heading over before Chamakh tested Szczesny with a long-range effort.
Both sides continued to trade chances in an open game, although Arsenal was aggrieved not to have been awarded a penalty as Chris Foy ruled that Damien Delaney's foul on Gnabry had been outside of the box.
Foy did point to the spot when Gnabry was brought down by Guedioura less than two minutes into the second half, enabling Arteta to step up and fire home confidently from 12 yards.
Wenger's men began to dictate proceedings following the opener and should have doubled the advantage shortly after, Giroud missing the target from point-blank range after good work down the right from Sagna.
However, the game was completely changed after 65 minutes when goalscorer Arteta was shown a straight red card for pulling down Chamakh as the Moroccan attempted to break through on goal.
Palace immediately began to take charge following Arteta's dismissal. Szczesny had to come up with some heroics to tip Joel Ward's fierce effort onto the crossbar before producing another flying save to keep out Mile Jedinak's volley.
But Giroud wrapped things up three minutes from time to end the home team's hopes of a late leveler.