The Spanish midfielder slotted home from the spot at the start of the second half when Serge Gnarby had been upended by Adlene Guedioura.
However, Arteta was then shown a straight red card for what referee Chris Foy deemed denying a clear goalscoring opportunity when he bundled over former Arsenal striker Marouane Chamakh some 35 yards out.
Palace, who parted company with Ian Holloway following the 4-1 home defeat by Fulham, rallied when Joel Ward saw his shot tipped on to the bar by Wojciech Szczesny, who then produced a wonder-save to deny Mile Jedinak from the corner.
Arsenal, though, secured all three points when Olivier Giroud headed in with three minutes left after a break by Aaron Ramsey down the left.
"We have a good ground to appeal. If he is only suspended for one game, I won't do it because he is suspended for Tuesday night (against Chelsea in the Capital One Cup) and anyway I would not have played him, but it was more today's game that was at stake that you worry about," said Wenger.
Arsenal were reduced to 10 men on 65 minutes, and then had to soak up a great deal of Palace pressure, when an equaliser would not have been unjust.
A long ball up field was taken on the chest by Chamakh, who looked to get clear before tangling with Arteta, shoulder to shoulder just inside the Arsenal half.
The pair went down in a heap, but that was enough to convince Foy the Gunners captain had denied his former team-mate a clear goalscoring opportunity and went straight for the red card.
"It was the wrong decision because it has to be a good goalscoring opportunity," Wenger said.
"He was far from goal, it was an accidental foul, he was not the last man.
"I think the referee got that one wrong and after that we dealt well with the situation.
"We were mature, organised and didn't become nervous and we got the second goal.
"Szczesny made two very good saves at an important moment of the game."
Arsenal finally secured three points when Ramsey broke down the left and delayed a chipped pass through for Giroud, who arrived on cue to nod the ball past Julian Speroni.
Wenger was pleased with the way his team closed out the match for a crucial victory following on from the disappointment of a midweek defeat at home in the Champions League to Borussia Dortmund.
"It was not a game of brilliance, but it was a game of efficiency and patience," he said.
"The pitch was a bit slow, maybe we were not the sharpest as well and the two together made us create less chances than we are used to, but we were serious and determined."
Wenger, though, is refusing to read too much into his side's solid start to the season.
"We are ambitious and want to do as well as we can, you can see that these players are focused," he said.
"Even when the legs go a little bit, they keep their legs and that is always a sign of ambition in the team.
"We want to win with style but unfortunately it is not always possible."
Palace caretaker manager Keith Millen, who made half-a-dozen changes to his side from Monday night's defeat, felt the team could be proud of their efforts.
"The last couple of days have been difficult obviously with Ian leaving, but from a work point of view it's been fantastic," he said.
"It was about playing for the fans and I think the fans will have appreciated the effort and commitment.
"Their response to going one down against a world-class side made me proud, and I am just disappointed they didn't get something from the game that I felt they deserved."
Former Stoke boss Tony Pulis has been linked with the Selhurst Park vacancy.
Millen added: "It is important that we get someone in really as quickly as possible, but it has to be the right person.
"I am sure the chairman won't rush in to that."
- Sports & Recreation
- Marouane Chamakh
- Wojciech Szczesny
- Olivier Giroud
- Ian Holloway