Phil Jones, Wayne Rooney and substitute Danny Welbeck ensured that Moyes travelled back to Manchester with all three points.
It has been a testing time for the United manager but he was delighted to return to wining ways and praised his team.
Moyes said: "I thought Robin van Persie and Wayne Rooney were excellent.
"Juan Mata and Adnan Januzaj too, but we also needed a good defensive performance as well so, all round, I'm pleased.
"The pitch was very lively, it was drying out a little bit and it wasn't that easy to pass the ball, but we looked much more potent going forward and looked more likely to score than we have in other games.
"I made the point about us trying to attack more and have better play going forward and I think we did that."
Chris Smalling impressed for England during the week and Moyes is hopeful that he will build a good understanding at the heart of United's defence with Phil Jones.
Moyes added: "I thought Smalling and Jones played really well.
"Smalling was excellent by all accounts for England too, while Jones is a very good player.
"We needed to make sure he's ready to play and [for the first goal] it was a really good header by Phil after Rafael did good work down the right and got the free-kick.
"Robin's delivery was excellent and, thankfully, Phil got across his defender and scored."
United entertain Liverpool at Old Trafford in the Premier League next Sunday, live on Sky Sports 1HD.