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 (right). "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."
City had been handed an early advantage when referee Mike Dean sent off Arsenal defender Koscielny for wrestling Edin Dzeko to the floor after just 10 minutes.
Although the Bosnian saw his penalty saved, via the post as the ball ran back along the goalline to Wojciech Szczesny, City -- who last won in the league at Arsenal in 1975 -- took command with first-half goals from James Milner and then a tap-in for Dzeko after Carlos Tevez's shot was parried.
While Arsenal were a tighter unit in the second half, Wenger felt his team had paid a heavy price for their poor approach.
"Overall we started too timid, with not enough authority in a game like that, and allowed them to dictate," he said.
"We didn't start with enough confidence or authority.
"It is frustrating because the team showed great heart and desire after that, but we were a bit too nervous to play in a serene way at home and that is costing us.
"It is not anger, it is frustration that we do not see from the start what this team is capable of."
Wenger maintained he had not yet reviewed Koscielny's red card, insisting that would not change the outcome.
"What do I make of the decision? That is not really important. You have to live with the decision," the Arsenal boss said.
"I was surprised, but honestly, I refuse to watch it again because what does it help now?
"I will watch it again. I have heard it was a penalty. Was it a red card? I don't know."
Abou Diaby came into midfield for what was a first appearance since September because of a thigh injury.
Wenger revealed Arteta could be set for a length spell on the sidelines.
"Of course (it is a blow). In midfield he gives us stability, experience and tactical knowledge. We will miss that," he said.
Wenger maintains Arsenal are "in the market" for reinforcements during the January transfer window, but will only move for quality men.
"We try very hard, believe me," he said.
"We have a team around me who analyse everything. We select targets, I watch them, but it is difficult to speak about any names."