Robin van Persie netted against his former employers in the third minute to give the hosts the perfect start, before Wayne Rooney missed a penalty just before half-time.
Despite dominating a subdued Arsenal, United failed to build on their lead until the 67th minute, when Patrice Evra headed home.
Arsenal midfielder Jack Wilshere was then sent off for a second bookable offence, before Santi Cazorla netted a consolation deep into stoppage time.
“I’m disappointed of course, because we had a lot of the ball but we were not very efficient in our final third or in their final third,” said Wenger.
“You feel it was a bit [of] an illusionary domination. We had a lot of the ball but they created more chances and deserved to win the game.
“Part of that is down to the fact that we gave an early goal away. In a game like that, that’s a massive advantage [for Manchester United].
“Then they defended well, were sharper than us in the duels and the consequence is you lose the game.”
Wenger, whose side are sixth in the Premier League table, insisted it made no difference that it was Van Persie who netted the opener.
“Honestly, that doesn’t bother me at all,” he said. “That doesn’t matter.
“What you want is to win the game. You don’t care about who scores in their team. What is most important is that you don’t make a mistake at the back and you don’t give chances away.”
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