One of the most dramatic finals of all time played in driving rain in Moscow ended with United winning 6-5 on penalties after Chelsea captain John Terry blew the chance of winning the European Cup when his penalty hit the post.
United then took their chance to win the shootout and become European champions for the third time.
Friday's draw also paired favourites Barcelona against Shakhtar Donetsk of Ukraine while nine-times winners Real Madrid will play Champions League debutants Tottenham Hotspur, the highest scorers left in this season's competition.
Holders Inter Milan, who squeezed into the last eight with an away goals win over Bayern Munich on Tuesday, will play Schalke 04, who beat them in the 1997 UEFA Cup final.
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Carlo Ancelotti, manager of English champions Chelsea, told reporters after his side were paired with United:
"It will be difficult, obviously, but we want to reach the semi-final. I'm not surprised two English teams have been drawn together, this can happen.
MOURINHO HAPPY
"For English football it's good because I think that compared to last season one team will be in the semi-final. Last season no English team was in the semi-final, this year one or two teams will be in the semi-final."
The other English team who could reach the last four are Spurs, who face Jose Mourinho's Real with the first leg in Madrid.
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Mourinho, who led Inter to a treble last season, including bringing them the European Cup for the first time since 1965, is now attempting to become the first man to lift the trophy with three different clubs after his success with Porto in 2004.
He said he was delighted to be going back to England to face Spurs.
"I am happy, not because it will be easy, but because I love playing in England," he said.
Spurs have been impressive in their debut season with wins over Inter Milan and AC Milan, who they eliminated in the last 16, and manager Harry Redknapp was also happy with the draw.
"What a game to look forward to against one of the greatest clubs in world football. To play Real Madrid at this stage of the Champions League is fantastic for us and we're all looking forward to it."
UNLUCKY DRAW
The club everyone wanted to avoid was Barcelona, the hot favourites to become European champions for the second time in three seasons.
Shakhtar were the unlucky side to be paired with Pep Guardiola's formidable outfit, but have won two of their six matches against them, including a 3-2 win at the Nou Camp in the group stage two seasons ago.
Guardiola told reporters: "We know them and they know us. We have had very tight games with them. They are physically very strong, with experienced players. We found it very hard to beat them in the Super Cup (in 2009 when Barca won 1-0 in extra time).
"Playing the return leg away is a handicap."
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