The Black Cats ousted Jose Mourinho's men with a come-from-behind extra-time victory at the Stadium of Light on Tuesday night and they will welcome United, who comfortably beat Stoke, in the first leg in the week commencing January 6 with the return clash at Old Trafford a fortnight later.
The other semi-final pits United's rivals Manchester City against West Ham, who earned their spot at this stage after beating managerless Tottenham 2-1 at White Hart Lane.
United last won this competition in the 2009/10 season and going all the way this season may go some way to easing the pressure on David Moyes, who has endured a frustrating start to his reign as Red Devils manager.
Following their 2-0 victory at Stoke on Wednesday night, the Scot admitted he is at a club accustomed to major honours, but is not looking past the struggling Black Cats, who are in the bottom three of the Barclays Premier League and whose best result in this competition was reaching the final in 1985 - a year they were relegated from the top-flight.
"We had to win the game before we got into the next round but thankfully we did and we'll have a tough semi-final against Sunderland now," Moyes told Sky Sports 1.
"It's a cup game and it shows you how well Sunderland can play when they beat Chelsea last night, they played well to get that result and they've got a new manager now.
"A lot of things can happen between now and then but we're just really pleased we've got ourselves into the semi-final.
"Manchester United Football Club are used to winning trophies and so it's no surprise to them. It's business as usual but I've got to say that I'm delighted that we're in the semi-final."
The draw kept the two Manchester clubs apart, with City, who beat Leicester 3-1 on Tuesday, hosting the Hammers in their first leg before travelling to Upton Park in the week commencing January 20.
City last won the competition in 1976, but they have been knocked out of the semi-finals in two of the previous four seasons, while West Ham's best result was a final appearance in 1981.