United were dominant from the outset and took the lead after just three minutes when ex-Gunners captain Robin van Persie pounced on a mistake by Thomas Vermaelen to open the scoring.
The Red Devils continued to churn out the chances with Wayne Rooney missing a penalty just before the break, while Antonio Valencia almost turned in a cross from Van Persie, who also forced two good saves from Vito Mannone.
Eventually, it was defender Patrice Evra who got United's second in the 67th minute when he headed home a cross from Rooney, and the strike proved to be vital as Santi Cazorla pulled one back for Arsenal in injury-time.
While it was his side that ultimately left Old Trafford with all three points, Ferguson believes they were far from at their best.
"It's unbelievable," he told Sky Sports. "The chances we missed, we really should have put it to bed a long time ago.
"You look at that scoreline, 2-1, you'd 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, we were far too casual.
"It was an unusual Manchester United-Arsenal game. I know there was a lot of bookings but it didn't really get anywhere near the Manchester United-Arsenal games of the past.
"It was a strange game. I need to sleep on it."
Ferguson was also impressed by the performance of Van Persie.
He said: "It was an ideal start for Manchester United of course, and it was good for Robin in particular.
"I thought he was fantastic today [Saturday], his movement and penetration were fantastic.
"I thought the back four did really well but after that, I'll just look forward to the next game."
Adding to Ferguson's frustration was the fact that his side did play almost 20 minutes with a numerical advantage after Jack Wilshere, who was booked in the first half and was fortunate to escape a second yellow card for a foul on Van Persie, was eventually being dismissed for bringing down Evra.
Fearing something similar happening to Tom Cleverley, who was also on a yellow card, Ferguson withdrew his midfielder.
The Scot said: "I think Jack Wilshere was warned for his third foul on Van Persie then Thomas did a foul on Wilshere and I think he got warned.
"I wasn't prepared to take a chance and Anderson actually changed the game a bit for us."