The Gunners lost more ground in their quest to break back into the top four, first failing to recover from the early sending off of defender Laurent Koscielny.
Arsenal could not get themselves back in contention when City were also reduced to 10 men for the final 15 minutes after captain Vincent Kompany was given his marching orders for a studs-up challenge on Jack Wilshere.
Arsenal - who must do without midfielder Mikel Arteta for up to a month with a calf injury - have to regroup quickly for an FA Cup third-round replay against Swansea on Wednesday night, when Wenger is expecting the required response.
"We have to fight to do as well as possible until the end of the season," said Wenger. "I feel we need to find in this kind of game more defensive stability.
"The way we gave the ball away, it was a combination of a lack of concentration and a bit of awareness."