The task facing Moyes' side was made that much harder after United were also drawn to play the second leg in Munich.
Jose Mourinho's Chelsea will face Paris St Germain, but will have the second leg at Stamford Bridge. Real Madrid will face Borussia Dortmund, while there is an all-Spanish meeting between Barcelona and Atletico Madrid.
Moyes admitted that Bayern were the tournament favourites, but insisted: "I was thrilled that we were in that draw and we're in the quarter-finals of the Champions League. 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 I'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."
Winning the tournament outright now looks like United's only chance of being part of Europe's elite club competition next season and missing out could cost the club around £50million.
It is understood, though, that United chiefs are confident that rises in commercial income will make up for the financial impact and allow Moyes and the club to attract the top players.
Pep Guardiola's Bayern have been in amazing form this season, unbeaten in the Bundesliga having won every match at home in that competition.
Moyes can take some solace from the fact that the only two clubs who have prevented Bayern from winning at home this season were both English. Manchester City won 3-2 at the Allianz Arena, while Arsenal achieved a 1-1 draw in the last round.
The two clubs were involved in that famous last-gasp victory for United in 1999 Champions League final, but Bayern took the honours last time they met, in the 2010 quarter-finals, when Arjen Robben's volley at Old Trafford made it 4-4 on aggregate to send the German side through on away goals.
Bayern director of sport Matthias Sammer said United's threat should not be under-estimated, even if they are struggling in the Premier League.
"It's a highly attractive draw," he told the club's website. "They showed in the second leg of their last-16 game what they're capable of.
"On a good day, they are deadly. We must ensure they don't have two good days. I'd say the draw's manageable, but dangerous."
Chelsea's draw against PSG will see Mourinho come up against his former striker Zlatan Ibrahimovic, whom he managed at Inter Milan, before the Swede was sold to Barcelona.
Ibrahimovic has spoken of his admiration for Mourinho, saying in his autobiography that he "was basically willing to die for" the Portuguese.
Chelsea chief executive Ron Gourlay said the club would treat PSG "with utmost respect".
He said: "Our form is good, we've got a lot of games between now and the end of the season. We're in a good position in the Premier League so we just need to take it one game at a time, try and win every game and see how everybody else does around us.
"We will treat Paris St Germain with the utmost respect and hopefully we can go there, put on a show for our fans and bring it back to Stamford Bridge."
The all-Spanish tie between Barcelona and Atletico should be a cracker - both sides are locked together in La Liga, with the Madrid side a point ahead in second place. The teams have already played out three draws this season and will face each other in the last league match of the campaign at the Nou Camp.
Real Madrid have the chance for revenge over Dortmund, who humbled them 4-1 in the first leg of their semi-final last season en route to a 4-3 aggregate victory.
Dortmund, however, have a number of significant injuries, while striker Robert Lewandowski is suspended for the first leg, making the Spanish league leaders clear favourites to progress.