Goals from Robin van Persie and Wayne Rooney settled the contest, and the boss was delighted to get up and running at Old Trafford while his predecessor Sir Alex Ferguson watched from the stands.
Moyes said: "I'm very happy. I thought there was a lot of good play, in the early part of the first half we were very good. We created a lot of crossing positions and goal-scoring opportunities which we didn’t take. We made it a little bit nervy, so I was delighted when we got the penalty kick before half time.
"I think he [Sir Alex] will be pleased, he knows the Premier League and he’ll have seen the team play well, create quite a lot of chances and do a professional job. We didn’t concede any goals, which is a great thing. At the moment, we’re looking strong defensively and that will be really important in the coming games."
United were awarded a controversial first-half penalty when Ashley Young appeared to be brought down just outside the area, but Moyes believes that a spot kick was no more than his men deserved.
He added: "I’ve had a chance to see it. There’s part of it I agree with, I think the first contact is outside the box, but I wasn’t so sure about the other one. The one on Patrice Evra could have been a penalty kick, we had two or three calls for it, but we deserved it because we played well enough and were the dominant team."