Sir Alex admitted the Reds needed a wake-up call before going on to negotiate a difficult FA Cup fourth round draw on Sunday.
Tottenham Hotspur scored first, but a Carlos Tevez strike and a Cristiano Ronaldo double gave United a 3-1 victory.
“I think it was a fantastic cup tie and both sides contributed to that,” Sir Alex told MUTV.
“It was quite an open match – we attacked and they also had some good chances. I think we also had great opportunities to create chances, although the final ball wasn’t so good at times today.
“I think [Robbie Keane’s] goal woke us up a bit. We were looking a bit slack at the time – we were too relaxed and there was no intensity in our game.
“But we controlled most of the possession and put a lot hard work in – we had to against what was a good Tottenham side.
“I’m very happy. We were playing a top-flight team and, remember, they came bouncing into the game today on the back of a fantastic result against Arsenal.”
Tottenham’s Michael Dawson was shown the red card on 68 minutes after handling the ball inside the area. Sir Alex admitted referee Peter Walton had little choice but to dismiss the 24-year-old, although sympathised with the Spurs defender.
“It was certainly a penalty kick but whether it was a sending off… maybe it was by the letter of the law.
“Referees are put in a difficult position with the mandate regarding the last man. I’d say [today’s decision] was harsh – there’s no doubt about that. I felt sorry for the lad [Dawson].
“You have to give Tottenham credit for keeping coming at us. They gambled by bringing Jermaine Defoe on and playing three up front. We seemed to sit back a little bit and when you stand back and give players of that ability time on the ball, you’re going to have a problem.”
Video: Watch Sir Alex's interview at MUTV Online.