Arsenal will go to Sheffield United in the last eight of the FA Cup, while Manchester City face a trip to Newcastle.
Should they beat Derby County on Thursday then Manchester United will go to Norwich, who beat Tottenham on penalties on Wednesday evening.
Leicester host Chelsea in the other tie.
Holders Manchester City made it through to the last eight with a 1-0 win over Championship side Sheffield Wednesday on Wednesday evening.
Following a goalless first half at Hillsborough, Sergio Aguero fired a largely dominant City in front in the second period when Wednesday goalkeeper Joe Wildsmith was unable to keep out the Argentinean's effort, which eventually dropped into the net.
Pep Guardiola's side pressed for the second goal throughout the second period, but were force to settle for Aguero's solitary strike thanks to some determined defending from Garry Monk's side.
Tottenham vs Norwich went to extra-time before the Canaries prevailed, Leicester battled their way into the last eight with a 1-0 victory over Championship side Birmingham City at the King Power Stadium, where right-back Ricardo Pereira headed the only goal of the game with eight minutes remaining.
Arsenal got the better of League One side Portsmouth with a 2-0 win at Fratton Park on Monday night, with Chelsea beating Premier League leaders Liverpool by the same scoreline at Stamford Bridge on Tuesday evening.
Sheffield United and Newcastle United also made it through on Tuesday thanks to away wins at Reading and West Brom respectively, with Derby vs Manchester United completing the fifth round line-up on Thursday evening.
The quarter-final draw in full:
Sheffield United vs Arsenal
Newcastle United vs Man City
Norwich vs Derby or Man Utd
Leicester vs Chelsea
The quarter-final ties will take place on the weekend of 21-22 March.
