Despite being 12 points adrift of a Champions League spot, the North-West outfit remain in this year's European competition following a second leg victory over Olympiacos on Wednesday.
"I think the big thing was I could look at the supporters in the face after the game on Wednesday night because I thought we had given them something to shout about, we had nothing to be ashamed of with our performance," the Daily Mail quotes Moyes as saying.
"In some other games I have felt ashamed because the team haven't played well enough and I've not got a team out which has got a good result.
"Our form hasn't been consistent enough, our performances haven't been consistent enough. We've obviously got to get better at that. It's going to take time, we're not going to be able to do everything immediately."
United will take on Champions League holders Bayern Munich in the quarter-finals.