Earlier in the day, the Reds had seen Roberto Mancini's men beat Tottenham with virtually the last kick of the game at the Etihad having been pulled back to 2-2 by Spurs.
The Blues' last-gasp victory took them six points ahead of United at kick-off at the Emirates, but an impressive display from the Reds and goals from Antonio Valencia and Danny Welbeck secured a crucial result.
"It was important to win after City had won their game, but the manner in which we won was the more pleasing thing for me," said Sir Alex afterwards.
"I think we won in the right way - we played really adventurous football, we were positive and had great belief in ourselves. I'm delighted to see that at this important time in the season.
"Arsenal came out fighting in the second half and for the first 15 minutes we rode our luck a bit. But on the counter-attack we were devastating and we could have scored a lot of goals.
"I said before the game that we had four massive away games [Arsenal, Chelsea, Tottenham and City] to come and that's one out the way.
"What we need to do is stay on City's coat-tails. Football is a funny game. The game at City today tells you things can happen and there will be changes [before the end of the season]. We've just got to stay on their coat-tails."
The manager reserved special praise for match-winner Danny Welbeck who netted his ninth goal of the campaign and his third in three games.