United overcame the league leaders following a first-half header from Robin van Persie.
"It's difficult to lose the game. We conceded a goal on a set-piece," Wenger told Sky Sports News after the match.
"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."
Despite the defeat, Arsenal remain top of the Premier League table with a two-point advantage over Liverpool.