John Terry famously missed his penalty in the shoot-out when he had the chance to win the trophy for the Blues.
The first leg between the Premier League rivals will take place at Stamford Bridge with the second at Old Trafford the following week.
"You have to consider the past," said Chelsea boss Carlo Ancelotti. "The players can be reminded they lost against Manchester United but also they can be reminded that we won last year [in the Premier League] and we won this season against them.
"The past is the past. They will be different games, very important games, very exciting. It will be difficult, obviously.
"For English football it's good because one [English] team will be in the semi-final. Last season no English team was in the semi-final, this year one team or two teams will be."
Chelsea secretary David Barnard said the Moscow defeat would act as a motivation for those players still in the team.
"We have a great rivalry domestically but the Champions League is something different," he said. "We're looking forward to it, people will bring up Moscow and revenge but we know each other well enough.
"Most of that team that experienced Moscow are still here so it'll be a great motivation. We've been the bridesmaid many times."
I'm happy not because it is easy but because I love to play in England
Real Madrid manager Jose Mourinho
Whoever does win will play the first leg of their semi-final against Inter or Schalke away with the return at home with a chance to make it to the final at Wembley on 28 May.
Javier Hernandez scored twice in United's win over Marseille in the last round and said: "Stamford Bridge is a nice stadium and it makes it a little bit like a league game. But if there is an advantage that we are at home in the second game."
United secretary John Alexander added: "It's a shame the English clubs were paired together. We have a good record against domestic clubs in Europe and I'm sure [manager] Sir Alex [Ferguson] will be prepared.
"It's a great chance for whoever comes through to make it to Wembley. Most teams wanted to avoid a clash with Barcelona."
Tottenham, who knocked out AC Milan in the last round, will travel to the Bernabeu for the first leg of their tie against nine-time winners Real Madrid.
Harry Redknapp excited by Real tie
"You want to play in these games," sais Spurs boss Harry Redknapp. "You want to manage in these games and the fans want to go and see these games. It couldn't be better.
"It's a fantastic 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.
"It's going to be a difficult game, obviously. They are a great side. But you have to look forward to games like this. These are the great days in your life and great days in Tottenham's history.
"We'll go there and give it a real go. It's an exciting game. You couldn't have a more exciting or tougher game.
"I've said all along that the two Spanish clubs were, for me, the favourites to win it.
"We've drawn Real Madrid who are pushing Barcelona all the way for the championship with big players and a fantastic manager. In my opinion, he's one of the all-time greats.
"But we've got a chance against anyone. We feel we can play against anyone and give them a game. We're looking forward to it."
The two clubs met in the quarter-finals of the 1984/85 Uefa Cup when the Spanish side beat Tottenham 1-0 away before a goalless draw at the Bernabeu helped them advance on the way to winning the trophy that season.
But Real have failed to win their last five Champions League encounters against English clubs and have lost four of those games.
Tottenham, who reached the European Cup semi-finals in 1962 when they lost to Benfica, will be out to stop Real manager Jose Mourinho's attempts to become the first man to win the Champions League with three different clubs.
He has guided Real to the last eight for the first time since 2004, with their last win in the Champions League coming in 2002.
Mourinho said: "I'm happy not because it is easy but because I love to play in England."
Real director Emilio Butragueno added: "We are very happy because over the last few seasons we have been unable to get to this round.
"This competition is the most prestigious in Europe. It is going to be a fantastic tie and Tottenham have great, great players, offensive players and are very competitive.
"I'm sure it's going to be very difficult for us. I'm hopeful our team will rise to the occasion."
Spurs signed playmaker Rafael van der Vaart from Real last summer and Butragueno said: "Rafael van der Vaart is a fantastic player and a great person and professional and I'm happy he's been very successful.
"I'm sure he'll strive to do a great game and help his team qualify for the semi-finals."
The tie between them promises goals as Tottenham and Real have scored the most goals in the competition of the remaining teams with 19 apiece, although the Spanish team have let in only three while Spurs have conceded 11.
If Spurs do progress, they have been drawn to be at home for the first leg of a semi-final against favourites Barcelona or last-eight debutants Shakhtar.
Real and United are the only unbeaten teams left in this season's competition while Inter have lost three games, which is the most of any of the clubs contesting the quarter-finals.
CHAMPIONS LEAGUE Q/F DRAW Tuesday, 5 April 2011 Inter Milan v Schalke Real Madrid v Tottenham
Wednesday, 6 April 2011 Barcelona v Shakhtar Donetsk Chelsea v Man Utd, QFL1, L1, 19:45
Tuesday, 12 April 2011 Man Utd v Chelsea Shakhtar Donetsk v Barcelona
Wednesday, 13 April 2011 Schalke 04 v Inter Milan Tottenham v Real Madrid
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.