Premier League side Watford beat Championship Leeds 1-0 thanks to Scott Wooton’s 53rd-minute own-goal.
But top-flight West Bromwich Albion were knocked out after second-tier Reading, beaten FA Cup semi-finalists last season, came from behind to win 3-1 at the Madejski Stadium.
Darren Fletcher put West Brom ahead in the 54th minute but Reading’s Paul McShane ensured their lead lasted just five minutes.
Royals defender Michael Hector headed in a free-kick 18 minutes from time before substitute Lucas Piazon made it 3-1 in stoppage time.
Everton, whose last major trophy came when they won the 1995 FA Cup, went into the quarter-finals after a 2-0 win away to fellow Premier League side Bournemouth in Saturday’s late kick-off.
Everton goalkeeper Joel Robles saved Charlie Daniels’s first-half penalty after James McCarthy handled inside his own box.
The visitors compounded Bournemouth’s agony when England midfielder Ross Barkley, with national manager Roy Hodgson in the stands at Dean Court, saw his deflected shot loop over Reading’s Australian goalkeeper Adam Federici in the 55th minute.
Romelu Lukaku made it 2-0 when the Belgium striker scored his 21st goal of the season 14 minutes from time.
Sunday sees Chelsea at home to fellow Premier League giants Manchester City.
Manuel Pellegrini, the City manager, has threatened to field a weakened team in response to what he sees as unfair fixture congestion after the FA bowed to demands from broadcasters to move the game from its original Saturday date.
City are away to Dynamo Kiev in the Champions League on Wednesday and then face Liverpool in the League Cup final at Wembley a week on Sunday.
“We must prioritise the Champions League, so we’ll see what is the best team we can put on the pitch for the FA Cup,” said Pellegrini.
But interim Chelsea manager Guus Hiddink was wary of a backlash from City, who come into the match on the back of successive defeats by Premier league title rivals Leicester and Tottenham.
“A big team always reacts. We are warned,” said Hiddink, who will be without injured captain John Terry.
High-flyers Tottenham Hotspur face Crystal Palace in another all-Premier League clash on Sunday, when West Ham are away to second-tier Blackburn.
Meanwhile Manchester United will look to ease the growing pressure on manager Louis van Gaal after their 2-1 Europa League defeat by Danish side FC Midtjylland when they face third tier Shrewsbury Town in the FA Cup on Monday.
