The Gunners failed in their quest to move into the top four despite taking the lead in controversial circumstances through Emile Smith Rowe in the 13th minute.
Bruno Fernandes equalised for the hosts on the cusp of halftime before Cristiano Ronaldo scored his 800th career goal to give United the lead seven minutes after the break.
Martin Odegaard equalised for the visitors shortly after but then turned villain by giving away a penalty which was converted by Ronaldo.
"It is what it is. When you concede three goals at Old Trafford you make it difficult to get a result," Arteta told Amazon Prime.
"I'm disappointed with how we conceded some of the goals. We had some really good periods, getting back in the game and scoring the second and then we throw it away. After we needed two goals to win it and we weren't able to do it.
"I think we've come a long way since the game at Anfield [a 4-0 loss to Liverpool] because we were able to maintain a high level for longer periods. We had a period after scoring where we just kept giving the ball away in dangerous areas and you'll get punished against this opponent.
"We had very good structure. We were a threat. They couldn't run because we had a good structure in our counter-press, but when it was 2-2 and we had to build our momentum to go and win the game we conceded a penalty.
"There were a lot of things I liked in the game. I really liked the behaviour of the team, individually some of the performances were really good. We had some really good moments in the game and when you score two at Old Trafford you expect to get something out of the game. That's a regret but it's our own fault.
"If you look at the game, you would say we deserved much more than what we got. But you get punished against this level of opponent, this level of players. You cannot do certain things that we did."
Arsenal are away to Everton in their next league game on Monday.