A 10th-minute red card for Laurent Koscielny for a last-man foul on Edin Dzeko gave the hosts an impossible task but in truth, they had been off the pace from the start and it was no surprise when James Milner and Dzeko scored before half-time.
Sixth-placed Arsenal are like chalk and cheese at the moment, thumping in goals from all angles in the 7-3 victory over Newcastle United last month but barely mustering a shot on target against the champions on Sunday.
"He [Arteta] has a scan on Monday, [he will be out for] approximately three weeks. It is a strain on his calf," Wenger told reporters.
"In midfield he gives us stability, experience and tactical knowledge. We will miss that."
Asked if he might try to replace Arteta in the January transfer window, Wenger added: "We are in the market. To find players of a calibre of Arteta, available in January, I wish you good luck. We try very hard, believe me. But it's difficult to speak about any names."
"What I make of the decision is not really important. I was surprised. Honestly. I refuse to watch it again because what does it help now?" the Frenchman said.
"Overall we started too timidly, with not enough authority in a game like that and we allowed them to dictate from the start."
England midfielder Jack Wilshere is improving vastly game by game after 17 months out injured and can do part of Arteta's job plus give Arsenal forward thrust, but he can not do it alone.
"He has a game that provokes tackles, he's incisive, he goes forward," he said.
