Senegalese striker Demba Ba settled the tie with a memorable goal early in the second half to send Chelsea into the last four for the sixth time in eight seasons.
Winners of four of the last six tournaments, Chelsea will play Manchester City in the semi-finals at Wembley Stadium on April 14.
Despite his unpopularity with the Chelsea fans, interim manager Rafael Benitez could yet end the season with two trophies, as Russians Rubin Kazan await in the Europa League quarter-finals on Thursday.
"It was a great win," said Chelsea goalkeeper Petr Cech, who produced a breath-taking save from Javier Hernandez to preserve his side's lead.
"We love Wembley, we respect the FA Cup as a whole and we always want to win.
"For my save, it was a great ball into the box and I just went as fast as I could and let my left arm do the job. Luckily I was able to reach it."
United, meanwhile, have now gone nine years without winning the FA Cup, having seen hopes of a league, cup and Champions League treble evaporate in the space of a month.
'We were caught napping'
Their 15-point lead in the Premier League means they are virtually guaranteed to succeed City as champions, but manager Alex Ferguson faces the odd prospect of an end to the campaign with nothing to play for.
"We spoke about Demba Ba, with the players' experience of playing against him when he played for Newcastle," said Ferguson.
"He scored that exact type of goal against us a while back and we were caught napping."
Benitez complained that Ferguson snubbed his offer of a handshake prior to the 2-2 draw in the sides' first meeting, but the United manager put the matter to bed by offering Benitez a perfunctory handshake as the Spaniard emerged from the tunnel.
Instead, the day's first headlines were created by Ferguson's team selection.
With Wayne Rooney absent due to a groin injury and Robin van Persie on the bench, Danny Welbeck supported Javier Hernandez in attack, while Phil Jones bolstered the midfield.
Benitez made six changes to the team beaten at Southampton on Saturday, and if there was unfamiliarity in the two line-ups, it told in a very cautious opening.
Neither goalkeeper was tested until United number one David de Gea blocked from Ba in the 31st minute, by which time the hosts had already lost Ashley Cole to an apparent hamstring injury.
Imprecision in attack
United's imprecision in attack forced them to rely on shots from range, with Nani drilling narrowly wide from 25 yards and Hernandez almost embarrassing Cech with a skidding shot that the Czech goalkeeper kicked away in unorthodox fashion.
The creative players were beginning to come into the game though, and at the other end Eden Hazard bent a shot over the bar from Oscar's lay-off.
Juan Mata had been quiet, but four minutes into the second half he created the opening goal with a lofted pass that Ba hooked past De Gea with a sublime volley.
A United equaliser seemed certain in the 61st minute when Hernandez ghosted in to meet Welbeck's cross with a close-range header, but Cech showed astonishing reflexes to deflect the ball over the bar.
Ferguson sent on van Persie, followed by Ryan Giggs, by which time Mata had threatened to extend the hosts' lead by thrashing a shot into the side netting.
United had another close shave moments later, with Hazard seizing on a loose pass from Michael Carrick and bearing down on goal, only to roll his shot wide from 12 yards.
Ramires, Mata and Oscar also shot wide, while van Persie sent both of United's best chances of an equaliser over the crossbar.
Mata could even have made it 2-0 in the 89th minute, but after a neat exchange with Hazard, his shot was fielded by De Gea. – Sapa-AFP