The Red Devils' boss was speaking just two days after his side had overturned a two-goal deficit to beat Olympiakos 3-0 at Old Trafford in the second leg, to book their spot in the last eight.
Moyes admitted: "I am thrilled that we were in that draw and the quarter-finals of the Champions League.
"After being 2-0 down after the away leg at Olympiakos, I don't think there were too many people putting us through to the next round.
"So, in a lot of ways, it didn't really matter who we drew but I've got to say we've probably got the favourites and they are the defending champions - so it's a really tough draw."
As for playing the first leg at Old Trafford, he added: "I always think you want to be at home in the second leg and that would be the preference.
"But the way the games are going now, counter-attack football, and teams taking a different approach to playing away, it matters less."
United's passage past the Greek champions came courtesy of a Robin van Persie hat-trick although the Dutch striker left the field on a stretcher with an injured knee.
As for Van Persie's injury Moyes said: "He needs another scan. It's not totally conclusive so we need to check him out.
"All I can say he won't be available for West Ham."
Moyes accepts he has little Champions League experience but is not unperturbed going into a match against the Bundesliga champions-elect next month, saying: "I've had help from a lot of people around us.
"And on Wednesday it meant I could look our supporters in the face after feeling ashamed after other games."
As for Bayern, they will arrive in Manchester firm favourites to advance to the semi-finals but sporting director Matthias Sammer is not underestimating their English opponents.
In their home league, Bayern are unbeaten in 25 fixtures, 23 of which they have won, and Sammer said: "It's a very attractive draw against an opponent who are fighting for stability in their championship as well as in the Champions League.
"Manchester United have shown in the past that they are capable of big things and we have to show them respect.
"On a good day, Manchester remain capable of playing at a very high level and it's up to us to make sure they don't have two good days. It's feasible."
Bayern skipper and Germany international Philipp Lahm added: "It's a super draw. We are the defending champions so obviously one of the big favourites.
"Manchester United are a super team even if they're having problems and we can't go into this blind. They have excellent individuals and we've had some great battles in the past.
"Playing the first match away from home will not be a major factor. We'll be going to Manchester to attack and score goals."
And winger Arjen Robben, who scored the winning goal in last year's final at Wembley, went on: "It's a good draw and we're playing the first leg away.
"Man U are not at their best and they're not at the top of the Premier League but that counts for nothing in the Champions League."
The first leg is on April 1st, with the return in Germany eight days later.