United controlled the game throughout but missed several opportunities, with the 70-year-old insisting the Red Devils should have won by a far greater margin.
"I was disappointed to be honest with you, game where we should have won by four or five," Sir Alex told reporters.
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"I think I need to sleep on it, I could not believe the chances we missed.
"We should have put it to bed a long, long time ago and you look at that scoreline 2-1 and you think it was a close game, but it wasn't a close game.
"I think there was a lack of urgency in our game throughout, far too casual. It was an unusual Manchester United - Arsenal game really.
"I know there were a lot of bookings, but it didn't get anywhere near the Manchester United - Arsenal games of the past, nowhere near it."
Jack Wilshere was sent off for two yellow cards, and Sir Alex admitted that he substituted Tom Cleverley in the second half, after he had been booked, to avoid having his midfielder dismissed.
He added: "I think Jack Wilshere had been warned for his third foul on Robin [van Persie] and then Tom had a foul on Wilshere and I think he got a warning from Mike Dean.
"I was not prepared to take a chance and Anderson actually changed the game a bit for us."