Barcelona will face David Beckham's side Paris St Germain - he is the only English player left in the competition - while Real Madrid will take on Galatasaray.
In the other two quarter-finals, Bayern Munich's reward for seeing off Arsenal is a meeting with Celtic's conquerors Juventus, while Borussia Dortmund will play Spanish side Malaga.
Unlike in previous seasons, UEFA did not also make the draw for the semi-final pairings at the same time - that will now take place next month.
Real Madrid head coach Jose Mourinho is aiming to become the first manager to win the Champions League with three different clubs, having achieved it previously with Porto and Inter Milan.
His side saw off Manchester United in the first knockout round and also qualified from a tough group comprising the domestic league champions of Spain, Germany, England and Holland.
Steve McManaman, the former Liverpool and Real midfielder who carried out the draw for UEFA, said: "Real have had to do it the hard way so far - they have beaten a lot of good teams."
Galatasaray's squad include two former Mourinho favourites - ex-Chelsea striker Didier Drogba and former Inter midfielder Wesley Sneijder.
Barcelona, who came from 2-0 down in the first leg of the last round to beat AC Milan 4-2 on aggregate, will come up against their former striker Zlatan Ibrahimovic.
PSG sporting director Leonardo said Barcelona were red-hot favourites, and played down speculation linking the French club with a move for Wayne Rooney.
Leonardo told Sky Sports News: "We are the opposite of the other clubs in these times of economic problems. All the big players - I am not talking about Rooney - want to come. It's something we are very happy with.
"Rooney is one of the best players in the world."
"We never did nothing, we never talked with nobody and if one day we decide to do something the first one who will know that will be Manchester United.
"It's not the case now, there is nothing, nothing, nothing.
"We have never moved for any other player, not only Rooney."
Leonardo said Barcelona were "red hot favourites", adding to www.uefa.com: "It is the most difficult draw against the best team in the competition. We play the return leg at their home ground. It won't be easy, but we are looking forward to it because Paris deserves to be part of what will be a celebration of football.
"We get the chance to measure ourselves against one of the greatest sides of all time. Barcelona are the red-hot favourites, it's a celebration in which anything can still happen".
Barcelona sporting director, and the club's former goalkeeper, Andoni Zubizarreta said the experience of being 2-0 down to AC Milan had hit home with the players.
He said: We have just experienced something in the round of 16 that demands us to be prudent. "We have to look on playing the second leg in Barcelona as being positive. PSG have very good players with a lot of talent and with their experience and knowledge of playing these types of games I am sure they will be challenging right until the end. They are very competitive and we will be coming up against several players we know quite well."
Borussia Dortmund coach Jurgen Klopp admitted his side and opponents Malaga had both been surprises.
"Borussia Dortmund and Malaga are two teams that have surprised with their performances in the Champions League this season," he said. "So one thing is certain: there will be a surprise team in the semi-final. The way they eliminated Porto shows that they have a lot of quality in their squad."
Malaga vs Borussia Dortmund
Real Madrid vs Galatasaray
PSG vs Barcelona
Juventus vs Bayern Munich
First-leg ties to be played on April 2 and 3, with the second legs to be played on April 9 and 10.
Tottenham appeared to be handed the easiest draw in the quarter-finals of the Europa League after being paired with Swiss side Basel.
Chelsea face a long trip to take on Russian side Rubin Kazan while Newcastle will play Benfica from Portugal.
In the other quarter-final, Fenerbahce from Turkey will play Italian side Lazio.
Rubin Kazan will be tough opponents for Rafa Benitez's Chelsea team.
The Russian side have already knocked out two Spanish teams - holders Atletico Madrid in the last 32 and Levante last night, winning 2-0 after extra-time in Moscow's Luzhniki Stadium.
Rubin were also unbeaten in a tough group that included Inter Milan, and are particularly strong defensively, having conceded only three goals in 10 European matches.
Spurs' opponents Basle scraped into the quarter-finals after Russian side Zenit St Petersburg missed a late penalty.
Newcastle's opponents Benfica came into the Europa League knock-out stages after finishing third in a Champions League group including Celtic and Barcelona, during which they achieved a 0-0 draw at the Nou Camp.
Former Newcastle and Holland striker Patrick Kluivert, who carried out the draw for UEFA, said he believed an English side has a great chance of winning the competition.
Kluivert said: "There are three great teams from England - Chelsea, Tottenham and Newcastle - and for sure one of those can lift the trophy but there will be other teams eager to do so too so that can only be good for the competition."
Chelsea vs Rubin Kazan
Tottenham Hotspur vs Basel
Fenerbahce vs Lazio
Benfica vs Newcastle
The first leg to be played on 4th April, with the second leg to be played on 11th April.