The one-time winner of the competition, lifting the cup at Wembley with the Blues in 2007, must now overcome Antonio Conte at his old Stamford Bridge hunting ground.
There is also a plum sixth-round draw for Lincoln City, the first non-league side to make it this far since 1914 thanks to victory over Burnley, who will potentially visit the Emirates Stadium should strong favourites Arsenal beat Sutton United on Monday evening.
Another side to pull off a big upset on fifth-round weekend was Millwall, whose reward for edging past Leicester City - their third Premier League scalp in a row - is a trip to face London rivals Tottenham Hotspur at White Hart Lane.
Quarter-final draw in full:
Chelsea vs. Man United
Middlesbrough vs. Huddersfield or Man City
Tottenham vs. Millwall
Sutton United or Arsenal vs. Lincoln City