The league leaders saw their 10-match unbeaten streak in away league matches – of which nine have been wins – come to an end at Old Trafford, as former player Robin van Persie scored the game's only goal.
The Dutchman's 27th-minute header, his third strike against Arsenal in as many matches since making the move, saw Arsenal miss the chance to reclaim their five-point lead at the summit.
Wenger remained upbeat despite the defeat – which followed impressive victories over Liverpool and Borussia Dortmund – but admitted his side were made to pay for their slow start.
"It's difficult to lose the game. We conceded a goal on a set-piece," Wenger told Sky Sports after seeing Arsenal's winless run at Old Trafford extended to 10 matches.
"In a big game, that made the difference. In the first half I felt we were too nervous and rushed our game too much.
"In the second half, when we played composed and had a real go and played our usual game, we were very, very, very, very close to scoring.
"We have to digest that and come back in a strong way. The heart and the desire (from the players) was in it, maybe a bit too much in the first half."
A sickness bug went through the Arsenal camp in the build-up to Sunday's fixture, with defender Per Mertesacker and midfielder Tomas Rosicky not involved in the squad.
Mikel Arteta and Serge Gnabry were also affected, but featured at Old Trafford, though the absence of Mertesacker robbed Arsenal of their tallest defender.
Van Persie exploited that to full effect, rising above striker Olivier Giroud to head in Wayne Rooney's corner, but Wenger refused to use the absence of the German – who was replaced by Thomas Vermaelen – as an excuse for the goal.
"We are quite strong recently on set pieces," Wenger said. "I was surprised that were caught on the near post.
"I believe Vermaelen had a very good game (though). He did lead the (defensive) line very well."
Wenger has now called on his players to hit back strongly after the international break, as they prepare to face the high-flying Southampton.
"(What am I looking for is) the response after the break," he added. "We had to give a lot this week.
"We played against Liverpool, Dortmund, here today, (but) I think a draw would have been a fair result today."
