A group of United fans had arranged for a plane to be flown over Old Trafford carrying a banner which read: 'Wrong One - Moyes Out', but the fans inside the stadium showed their support for the manager.
A difficult day for Moyes got worse when Ashley Westwood opened the scoring in the 13th minute, but Wayne Rooney scored twice in the first half to turn the game in United's favour.
Juan Mata then netted his first goal for the club and Javier Hernandez added a late fourth, to leave Moyes delighted to claim three points after the disappointing derby defeat in midweek.
"We didn't get off to a great start," Moyes told MUTV. "I think everyone was a bit nervy in that first five to 10 minute period. Once we regrouped and got going, it was good that we got the equaliser and we scored the second goal at a really good time just before half-time.
"I thought the players took the challenge on and rose to it after that and we got better.
"I love my football and the fourth one was a great goal with the interplay on the wing between Adnan Januzaj and Marouane Fellaini and Adnan put in a great cross for Chicharito. So it was a goal for a football supporter."
Moyes received warm applause from fans as he emerged from the tunnel before his players, and there were also boos from the crowd as the plane flew overhead.
"I just wanted everybody to know I'm the manager of the club," said Moyes. "They know where I am. I stand by the side of the pitch. Ever since I've been at Manchester United, I've only had support.
"I believe that all Manchester United supporters are behind me and they all understand the situation we are in. The players are incredibly focused, that was an important thing as I wanted the focus to make sure it was right."
United face Bayern Munich on Tuesday in the first leg of their UEFA Champions League quarter-final, and Moyes is sweating on the fitness of Rafael after having to bring on Michael Carrick for the Brazilian at half-time and rearrange his defence.
"With Rafael, it's an injury and we need to check it out," revealed Moyes. "Valencia had a bit of tendinitis in his knee, which is why he wasn't involved.
"He was probably one I would have put to full-back if we had lost one, so we knew we'd be short if that was the case."