It looked like United's search for a win would continue but an energetic second-half display, which included goals from Antonio Valencia Danny Welbeck, ensured the home side went away with all three points.
And, speaking to Sky Sports after the game, Moyes thought his side were worthy winners.
Moyes said: "We thought we played well. We started well but didn't do enough to get enough goals. Danny had a great chance to make it 1-0 early on but didn't.
"We limited them to a few opportunities and we got the result we deserved."
Moyes swapped Adnan Januzaj and Shinji Kagawa at the break and it was a change that turned the game in United's favour.
Moyes added: "We moved Januzaj wide and I thought that worked much better especially with Shinji off the front man.
"Adnan is really earning his start at the moment.
"We also had to work harder and make more chances. They went about their business really well.
"Quite often recently we've been missing that bit of luck to get someone on the end of chances but today we had it.
"Welbeck looked better in the second half too and was a threat. He could have scored in the first half but I am delighted he got his goal and we should have scored more through Chris Smalling and Kagawa but we'll maybe keep the goals for later in the season."
Moyes takes his side to Stamford Bridge next weekend to face Jose Mourinho's Chelsea.