Two second half goals from Wayne Rooney handed the home side three points, with Luis Suarez's late strike proving only a consolation for the Reds.
"It was a great performance by us – I'm disappointed it was only 2-1, to be honest with you," Sir Alex told Sky Sports.
"Maybe we did tend to just try and play out the game and keep possession and not really threaten in the way we did before we got the goals.
"But that's the only fault I can find of the players, because right through the team they played some fantastic, really good football today."
The 70-year-old did admit, however, that Liverpool's late rally, including a goal for Luis Suarez, made for a nervous finale.
"My heart's still out there," he admitted. "It was nervy in the sense that you're playing Liverpool, and they've come back from 2-0 down to 2-1, and you know football – anything can happen.
"You only need a second to score a goal or lose a goal. So that was a worrying period, but we've got through it."
Sir Alex did not think the introduction of Andy Carroll, Craig Bellamy and Charlie Adam from the bench made too much difference in the flow of the game, though.
"They went 4-4-2, and so had more focus on playing the ball forwards, whereas with the one striker in Suarez, they had to build around the game before they could get the ball to him," he said. "That was the only difference really."
On the balance of the whole match, Sir Alex felt his team fully deserved their victory and added that Wayne Rooney was not their only goal threat.
"I thought Danny Welbeck was terrific, particularly in the first half," he said. “Paul Scholes was unlucky with his header, as the goalkeeper's made a good save.
"I thought we were well on top of the game, and it was just about getting that goal. Wayne, of course, has scored and then two minutes later scored another one, which was a great start to the second half."
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