Manchester City face Tottenham Hotspur in an all-Premier League clash, their first-ever European meeting, while Cristiano Ronaldo's Juventus meet Ajax Amsterdam.
Barcelona will meet Manchester United in the Champions League last eight in a repeat of the 2009 and 2011 finals which were both won by the Catalans, with the help of goals from Lionel Messi.
Manchester City face Tottenham Hotspur in an all-Premier League clash, their first European meeting, while Cristiano Ronaldo's Juventus meet Ajax Amsterdam, who are back in the last eight after a 16-year absence.
Five-times European champions Liverpool, the fourth English representatives and beaten finalists last year, will face Porto in a repeat of last year's last-16 tie which they won 5-0 on aggregate.
The semi-final pairings were also made with Tottenham or City to face Ajax or Juventus and Manchester United or Barcelona to meet either Liverpool or Porto.
The draw had an unusual feel with neither Bayern Munich nor Real Madrid in the hat, the first time both European giants have failed to reach the last eight since 2005-06.
Five-times winners Barcelona, who are in the last eight for the 12th successive season, will be away in the first leg after UEFA switched the order to avoid United's home match clashing with Manchester City, who host Spurs in their second leg.
Tottenham and City will meet three times in 10 days in April with a Premier League fixture following hot on the heels of their second leg.
"We have already proved last season against Real Madrid and Dortmund, this season against Inter or Barca away, that we are ready to compete," Tottenham goalkeeper Hugo Lloris said. "Then you need a little luck and team spirit."
Ajax, once a major European power who struggle to compete financially with clubs from bigger leagues, ousted Real Madrid in the last 16 and their tie against Juventus will be a repeat of the 1996 final, which the Serie A club won on penalties.
They also met in the final of the old European Cup in 1973 with the Dutch side winning 1-0.
"I'm neither happy nor sad. Ajax eliminated Madrid so they are no joke," Juventus director Pavel Nedved said. "I liked them a lot against Madrid so we will have to be very careful. We will have to face Ajax with great commitment. I expect two great games."
Juventus are unbeaten in their last 10 games against Ajax, winning five of the last six.
Porto, who won the competition in 2004 under Jose Mourinho, are confident they can spring an upset after last year's mauling.
"We have to keep dreaming and believe we can go further," said midfielder Jesus Corona.
"What can I say?" said Liverpool manager Juergen Klopp, whose side knocked out Bayern in the round of 16. "It's Porto, we've played there already and we know how good we had to be last year. We saw the character and quality Porto have."
Chelsea were the last English team to win the Champion League in 2012.
The two-legged quarter-finals take place on April 9-10 and 16-17.
Napoli face Arsenal, outsiders Slavia meet Chelsea
Napoli, chasing their first European title since Diego Maradona's team won the old UEFA Cup in 1989, were drawn against Arsenal in the quarter-finals of the Europa League on Friday.
Rank outsiders Slavia Prague, who ousted Sevilla in thrilling style on Thursday, were rewarded with a tie against Chelsea, the other English Premier League team in the last eight.
Villarreal and Valencia were drawn together in an all-Spanish tie and Benfica will meet Eintracht Frankfurt, who eliminated Inter Milan with a 1-0 win at San Siro.
Arsenal were initially drawn at home in the second leg against Napoli but the order was reversed by UEFA to avoid a clash with Chelsea, who host Slavia in the return of their tie in London.
Arsenal or Napoli will meet Villarreal or Valencia in the semi-finals while Benfica or Eintracht play Slavia or Chelsea.