The task facing David Moyes' side will be even harder after they 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.
The Champions League is realistically the only trophy still available to Moyes' men but Pep Guardiola's Bayern have been in amazing form this season, unbeaten in the Bundesliga and 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 actually 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 the 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.
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.
The all-Spanish tie between Barcelona and Atletico should be a cracker - both sides are locked together in the Primera 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 Borussia Dortmund, who - thanks to four goals from Robert Lewandowski - humbled them 4-1 in the first leg of the semi-finals last season en route to a 4-3 aggregate victory.
Dortmund, however, have a number of significant injuries which should make the Spanish league leaders clear favourites to progress.
Bayern's director of sport Matthias Sammer said United's threat should not be underestimated 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 chief executive Ron Gourlay said the club would treat PSG "with utmost respect".
He said: "It was always going to be tough when you looked at the teams who were left in the draw. Paris St Germain is a very tough game but we'll be ready for the first match.
"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 to 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."
PSG president Nasser Al Khelaifi added: "All seven teams left (that PSG could have faced) are very strong. We will be ready for the games against Chelsea, a club which I have a lot of respect for. I know that it will be very exciting for all our fans and we are very much looking forward to it."
Dortmund coach Jurgen Klopp insisted he would "not be hoisting the white flag" against Real Madrid and the German side does have the benefit of the second leg at home.
"We're really looking forward to this challenge and we'll try to take the small chance that we've got," he said on his club's website.
"It's a good draw. There's going to be a cool atmosphere for both legs."
The draw for the quarter-finals is as follows:
Barcelona v Atletico Madrid
Real Madrid v Borussia Dortmund
Paris St Germain v Chelsea
Manchester United v Bayern Munich
First-leg ties to be played on April 1 and 2, with the second legs being played on April 8 and 9.