United's first season under the Scot has been inconsistent, and was compounded by their 3-0 defeat to Liverpool in the Premier League last weekend.
And Moyes admitted their 3-0 victory over Olympiacos in the UEFA Champions League on Wednesday, that secured a 3-2 aggregate win, was the first step in repaying the supporters.
"I think the big thing was I could look the supporters in the face after the game on Wednesday night, because I thought we had given them something to shout about," he said at a press conference on Friday.
"We had nothing to be ashamed of in our performance.
"In some other games I felt ashamed when I felt the team hadn't played well enough, I've not got a team out that's not got a good enough result."
After reaching the last eight of the competition at the expense of their Greek opponents, United were drawn against the favourites and defending champions Bayern Munich.
Moyes is relishing the chance to test themselves against the Bundesliga leaders.
"I was thrilled that we were in that draw and we're in the quarter-finals of the Champions League," he said.
"When we were 2-0 down to Olympiacos a week or so ago, I don't think a lot of people were putting us through to the quarter-final draw. So, in a lot of ways, it didn't matter who I drew.
"But I've got to say we've drawn out the favourites and obviously the holders of the cup competition as well, so it's a really tough draw for us but one that I'm looking forward to.
"I think if you're old fashioned, you'd say you always want to be at home in the second leg (United are at home first). I've got to say that would be the preference.
"But the way the game's going now with counter-attacking football and a different approach by teams away from home, we've got it and that's the way the draw came out.
"We're at home in the first leg and away in the second leg. We just have to take it."
