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.
Bob the manc, The real United fans have been standing beside Moyes since he came in and despite the poor results I can see that he will build a great dynasty at United if given the time. He is exactly the type of manager we need, someone brave to build his own vision at the club, not someone to just paper over the cracks and squeeze as much is possible from an already made team. It is time consuming work and he was destined to have a difficult start, the media and everyone apart from United fans want us to fail, any single result against United was always going to draw a lot of attention. I have been impressed by Moyes's emphasis on youth and his enthusiasm to build his own team. Hopefully he does what Sir Alex should have done years ago, rid us of under performing stars like Anderson, Young, etc. and get in new blood from the academy or the transfer window. Once he builds his own team, playing the 'United' way will come, this season however is all about reaching our targets. The imminent return of Van Persie and Rooney will help us, we have missed their interplay upfront in quite a few games. Jones has been missed as well, and until recently so was Carrick, if anyone thinks Sir Alex wouldn't have struggled with such injuries they are deluding themselves to make a case against Moyes.- utd_till_i_die