The Londoners pinned United back inside their own half for long spells at the Emirates Stadium, only for Louis van Gaal's side to lead against the run of play through Kieran Gibbs' 56th-minute own goal.
Wayne Rooney added a second on the break before the returning Olivier Giroud pulled a goal back in stoppage time.
"It's a game that we dominated for 80 per cent of the time and it's a long time since we dominated games against Man United like we did today," said Wenger, whose eighth-place side are 15 points behind leaders Chelsea after 12 games.
"But at the end of the day we were not efficient enough in the final third offensively and made a mistake at the back and they took advantage of it.
"We had plenty of chances, their keeper [David de Gea] was man of the match but that was the story of the game.
"We played with freedom and we pressed them very high. We won balls back and had plenty of opportunities to finish this game off.
"We made a big mistake on the first goal and when we were 1-0 down we were too impatient, were not cautious enough and we gave a second goal away."
Wenger's misery was compounded by goalkeeper Wojciech Szczesny and midfielder Jack Wilshere leaving the field through injury.
Szczesny was injured when he collided with Gibbs in failing to collect the cross that led to United's opener.
Wenger does not believe the Poland international is seriously hurt but Wilshere, who has a long history of ankle problems, is a cause for greater concern after he went down under a robust challenge from United defender Paddy McNair.
"Wojciech is not bad; Jack, I don’t know how bad it is because he tried to stay on and he couldn't," Wenger added.
"It's an ankle problem - I don’t know honestly how bad it is."
