After a quiet opening 45 minutes, Demba Ba opened the scoring three minutes after the break with a superbly executed volley into the top corner.
Robin van Persie wasted two opportunities near the end of the match to see Chelsea win 1-0 and set up a clash with Manchester City on April 14.
Here, Sports Mole takes a look at how the action transpired at Stamford Bridge.
On target 8
On target 3
Was the result fair?
The match could have gone either way, but Manchester United cannot argue about the result. The game was in the balance before Demba Ba's opening goal, but the visitors created at least three significant chances to equalise. Chelsea could have extended their advantage, and even though they failed to do so, they deserve to be in the semi-finals.
Their display in the first 45 minutes bore a resemblance to that of their performance against Southampton on Saturday, but they were much improved after the break. The goal set them on their way, but given their fixture schedule, their application in the second half must be applauded and they did what they needed to do to see out the game.
Manchester United's performance
Like Chelsea, Manchester United failed to perform in the first half before improving in the second. They put the Chelsea defence under severe pressure at times and were unlucky not to equalise through Javier Hernandez. If the Mexican forward had scored, then it could have changed the outcome of the match, but he didn't and United can have no complaints about the result.
Sports Mole's man of the match
Demba Ba: The Senegal striker was on the periphery of the game for large parts of the first half, but his excellent finish after 48 minutes sparked an improvement in his performance. His volley deserved to settle any game, but his hold-up play and running of the channels earns him the man of the match award.
Apart from a couple of wrong decisions when it came to awarding corners, Phil Dowd didn't have much impact on the game, which is what you want to see. None of the cautions that he handed out can be deemed harsh, and this was a competent display from the official.
Chelsea: Chelsea are back in action in just three days' time when they host Rubin Kazan in the Europa League. Interim manager Rafael Benitez will continue to rotate his squad and supporters of the Stamford Bridge outfit can expect to see a return for the likes of John Terry, Frank Lampard and Fernando Torres.
Manchester United: Manchester United have a week-long break before they face Manchester City. A win against their rivals, and the Premier League title race will be close to being decided.