Javier Hernandez scored the only goal soon after the break, losing his marker Jose Enrique in the box before slotting home Wayne Rooney's corner.
The win was the perfect response to United's 4-1 mauling by neighbours City on Sunday and Moyes was delighted to pick up the victory against a Liverpool side who had beaten the Red Devils 1-0 at Anfield in the league.
He told Sky Sports 1: "I thought it was a really good game. I thought both teams went for it, Liverpool had one or two chances, we did as well.
"Both games have been tight and have been sealed by a set-piece, quite similar in a lot of ways, and that was the way we got it tonight.
"(Hernandez's) movement was great. It's sort of real centre forward movement and I think that's what Chicharito's done in all his goals, every time they tell me about him, he's great in the box, he gets his goals.
"And not only his movement, the way he finished it, he had to adjust his body to get a finish away and it was a great time to score the goal, just after half time."
He added: "We lost at Anfield a few weeks ago, we lost on Sunday at the Etihad so we had to bounce back, we had to get some good results and I thought it was a really good result tonight.
"It gave me a chance to see a lot of different players playing who've maybe not played as much as they would have liked so it's given me some food for thought.
"I thought some of the performances were really good."
United reward for winning is a fourth round home tie with Norwich.
Moyes added: "You really only want home draws if you can get it. Norwich are a good side with a good manager and we'll pay them every respect."