The Blues won 1-0 thanks to Kiernan Dewsbury-Hall's strike, with the visitors also playing for the majority game with ten men after Idrissa Gueye's dismissal for clashing with teammate Michael Keane.
Moyes later remarked: "Look, it's a night for the supporters – but a long night for them because for 80 minutes we had to put up a wonderful defensive show and get a result and, oh boy, didn't we have to do it. Look, we started well, we played well in the game. Even in the first half when we went to 10 men, I thought we put in a good effort to try to keep the ball and make opportunities.
"The second half became much more difficult, which we expected. The keeper has to make a couple of saves, but the resilience and the determination, commitment amongst the players is great. We had an incident tonight, but sometimes it's not a bad thing that the players demand off each other, but unfortunately it led to a red card."
Moyes continued: "I think we were the better side at that time and we showed a lot of really good things on the ball, trying to create chances and play better. If you concede with 10 men, you may go on to lose the game so it was so important we didn't allow them one goal. It's another clean sheet as well.
"The goalkeeper, the defenders, it's great. We had a clean sheet against Fulham in the last game. We've come and won here at Old Trafford, which has not been an easy place to come and win. And we're playing against an improving Manchester United team as well."
No-one expected red card
On Gueye's red card, Moyes admitted he remained baffled by the decision.
"I couldn't understand it because I don't think anybody would have expected that. Supposedly, Anthony Taylor told me, by the letter of the law, if you raise your arm then you could get sent off and I get that – but I don't think there was anyone else in the stadium who would have thought there was anything in it.
"I think everyone was looking at it and thinking it was two players getting together and, if you want to know, I quite like it when my players have a bit of a fight with each other because it shows they care, it shows they want to do better and it shows the demands on players. If we hadn't have had that in the game tonight for long periods – demands and making sure we put each other under pressure to do the right thing – we might not have got a result.
"I'm disappointed Gana has got sent off, he's apologised to the players in the dressing room and to Michael and to everybody involved in it, and he praised the team greatly for the performance and the result tonight. We move on."