Robin van Persie put the hosts in front after just three minutes, but United had to wait until the 67th before a second goal arrived via Patrice Evra.
Numerous chances had gone begging before Evra found the net, including a Wayne Rooney penalty which he fired wide, and it's something that is worrying Sir Alex.
"It looked as if it was going to be self destruction," the manager told MUTV and Sky Sports. "It only takes a second to lose a goal.
"They had some possession without really threatening us, but thank god we got the second one because the fact they scored right on full-time would have been an embarrassment [if we'd only have been 1-0 up].
"It was a strange game and the chances we missed... well, we should have put the game to bed a long, long time ago. If you see the scoreline you'd think it was a close game, but it wasn't.
"There was a lack of urgency in our game throughout. We were far too casual. It was an unusual Manchester United/Arsenal game. There were a lot of bookings but it never really got anywhere near the Manchester United/Arsenal games of the past.
"I was just really disappointed in our ability to hammer home the advantage," he added.
"I spoke about goal difference to the players at the start of the season. We lost [the title] on goal difference last season and I don't want to