The England international's effort also gave United a crucial away goal, but boss Sir Alex Ferguson was forced into two alterations at Old Trafford, with Nani and John O'Shea replacing Antonio Valencia and Rafael.
As for Chelsea, Carlo Ancelotti opted to place faith in Fernando Torres, who was still looking for his first goal for the Blues since a £50m move from Liverpool, while Alex was given his first start since November.
It was a high-octane start from the home side who immediately had Chelsea on the back foot, but for all their pressure, headed by the hard-working Park Ji-Sung, United couldn't test Blues stopper Petr Cech.
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After a shaky opening, the visitors were beginning to dominate proceedings, but after Anelka had again gone close, United responded by putting the ball in the net at the other end, only for Javier Hernandez's close-range strike to be ruled out for offside.
The tension inside Old Trafford was being passed over to the players as the referee dished out three yellow cards in the space of a few minutes, but the home crowd were on their feet two minutes before the break when Hernandez put their team in control of the tie.
After being awarded a free kick in a wide position, Ryan Giggs delivered a perfectly flighted cross for the Mexican to convert the ball from two yards out to leave Chelsea with a tough ask after the half-time interval.
Hernandez's goal forced Ancelotti to act, and the Italian chief decided to withdraw Torres for Didier Drogba, and the Ivorian almost made an instant impact when he turned Rio Ferdinand before driving the ball marginally wide of the far post.
The Blues looked a greater threat after the introduction of Drogba, but United were happy to hit Chelsea on the break at will, with Nani's long-range shot being palmed away by Cech.
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With 20 minutes remaining in the contest at the Theatre of Dreams, Chelsea's hopes took a significant hit when they were reduced to 10 men after Ramires was handed a second yellow card for a foul on Nani.
However, five minutes later, the Blues found an unlikely equaliser through Drogba, who reacted fastest to Michael Essien's cross to volley the ball past Van der Sar to leave the visitors a goal away from the semi-finals.
That hope was short-lived, though, as United immediately went down the other end to put the tie to bed. Giggs, the creator of the first goal, found Park on the left-hand side of the penalty area, and the South Korean made no mistake in driving the ball home to eliminate Chelsea from the competition.
United's triumph set up a last-four showdown with Schalke 04, but after they breezed past the German outfit 6-1 on aggregate, they came undone in the final against Barcelona, falling to a 3-1 reverse at Wembley.
Chelsea's defeat left them to concentrate on the Premier League, and although they couldn't secure the league title after failing to record a win in their final three fixtures, they held off the challenge of Manchester City to secure the runner-up spot.