Moyes has had a torrid opening season in the Old Trafford hot seat but United's 3-0 win on Wednesday, courtesy of Robin van Persie's hat-trick, overturned a 2-0 first leg deficit to set up a quarter-final clash with holders Bayern Munich.
That came just three days after United's disappointing season plumbed new depths with a 3-0 home defeat by arch-rivals Liverpool in the Premier league - a result that raised serious questions about Moyes's position.
"In some of the games I have felt ashamed because we haven't played well enough and I have not got a team out that has got a good enough result," Moyes told a news conference on Friday.
"I think the big thing on Wednesday was that I could look at the supporters in the face after the game.
"We had given them something to shout about. There was nothing to be ashamed of about our performance."
With a top-four finish in the Premier League now looking almost impossible for United who are 12 points behind fourth-placed Manchester City, winning the Champions League is their only realistic hope of qualifying for it next season.
United are the least favoured of the eight teams left in the draw but Moyes said he was just glad to be still in the mix.
"In a lot of ways it didn't matter who I drew but I've probably drawn out the favourites," said Moyes, whose side will be at home in the first leg next month.
"They are the holders as well, so it's a tough draw but one I'm looking forward to. I've got us to the quarter-final and now the aim is to get to the semi-final.
"If you're old-fashioned you'll probably say you want to be at home in the second leg and that would be the preference but we'll just have to take it."
United will look to build on their win over Olympiakos at West Ham United on Saturday when they will be without striker Van Persie who injured his knee on Wednesday and was undergoing another scan on Friday to assess the damage.
"There is no guarantee that one good result will lead to another," Moyes said. "But you would rather have a good one than a bad one and we got a really good one in midweek.
"There have been periods this season when we've gone five or six games unbeaten and then we've drawn the next game so there have been periods when we've done it."
After the trip to West Ham United have a home derby against title-chasing Manchester City on Tuesday.
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