David Moyes' side produced an impressive, and crucial, 3-0 win over Olympiakos in their last-16 second leg on Wednesday and have been rewarded with a two-legged tie with the holders.
The first leg will be held at Old Trafford.
Meanwhile, Chelsea travel to Parc des Princes to face Ligue 1 champions PSG before the return leg at Stamford Bridge.
The two teams have faced each other once in the past, coming in the 2004-05 group stage as the Blues beat the French side 3-0 before being held to a 0-0 draw at home.
Elsewhere, two of La Liga's title-chasing sides go head to head as Barcelona and Atletico Madrid meet for the fourth and fifth time this season - the previous three meetings have ended as draws.
Real Madrid and Borussia Dortmund complete the quarter-final draw in a repeat of last season's semi-finals which the German side won 4-3 on aggregate.
The first legs will be played on April 1 or 2 before the second legs on April 8 or 9.
|THE QUARTER-FINAL DRAW
|Real Madrid||vs||Borussia Dortmund|
|Manchester United||vs||Bayern Munich|