Two goals from Samir Nasri proved sufficient with United's last minute reply proving of no consequence.
The win lifts the Gunners above United in the table.
Gunners boss Arsene Wenger admitted that it was a vital win.
He told Sky Sports: "I can take criticism. I am in a job where I have to accept that. There is only one answer to give, and that is the quality on the football pitch.
"It was of course vital to win. Overall, it was a great football game. Both teams really went for it.
"I knew today that was a big game for the future of the team. We needed to keep in the race and know we can win big games.
"Man United are a great side and they created chances but didn't take them. We took our chances today.
"They took the initiative when they were 2-0 down. They scored very late and we had a nervy six minutes in the end.
"We continued to go forward. When you play football in a negative way you put yourself under pressure. We tried to score the third goal.
"We are player-ish, you know. That always takes over. When you are 19 or 20, you want to play. But overall, it was a great performance from my team."