Antonio Valencia and Jesse Lingard gave a 2-0 advantage to United inside 11 minutes after punishing defensive mistakes by Laurent Koscielny and Shkodran Mustafi, respectively.
Arsenal, however, managed to bounce back and Alexandre Lacazette gave them hope of securing at least a point from the game after halving the deficit early in the second-half.
However, United goalkeeper David De Gea produced a Man of the Match performance to deny Arsenal the equaliser before Lingard scored again to make it 3-1.
Wenger believes that Arsenal did enough to even win the game, but the boss said that the lack of efficiency and their defensive mistakes left them with "nothing to show for" it.
"I think the quality of our performance was good but we were not decisive enough. I believe at the back, we didn't start well at all and we put ourselves in a very difficult position," Wenger said following Arsenal's 3-1 defeat to the Red Devils.
"Despite that, there were enough chances to come back and I don't know if you have watched many top-level games, but when you have 33 shots on goal in a game of that stature, it means that our dominance was top class. I'm angry and disappointed because at the end, we have nothing to show for it and even at 2-1, we made another mistake at the back that made the game even more difficult."
Arsenal eventually ended the game with 32 shots with the expected goal metric suggesting that the Gunners could have eventually won 5-2.
Asked about that, Wenger added, "Yes, of course [that's clear] if you have seen the game. What would be fair in football is that in these kinds of games, you have to be efficient and we had the chances to score, we didn't take them and their keeper as well had an outstanding game. In these kind of games, from the start, you have to dictate your strengths in the duels from the back and I felt that at the back, our concentration wasn't sharp enough from the start and not dominant enough.
"We have produced an excellent performance, but we conceded three goals and we had at least 10 chances and we were not efficient enough in the box. That's the reality of that and I feel sorry for the players because our spirit was outstanding, the quality of our performance was outstanding, but at the end of the day, we weren't efficient enough because we didn't start enough in areas where it matters."
Furthermore, Arsenal were hit by a further blow during the game with Mustafi being forced off after only 11 minutes into the game.
Wenger replaced the German international with Alex Iwobi, but when asked about the defender's injury, he said: "I don't know what [injury] he has. I don't know what's wrong with him at all."