Manchester United crushed Derby County 4-1 away while Everton beat Aston Villa 3-1 at Goodison Park on Sunday to reach the quarter-finals of the FA Cup.
United, winners of the competition a record 11 times, will visit either Swansea City or Fulham in the last eight, while Everton will be at home to West Ham United or Middlesbrough.
Chelsea, the only other team definitely through after beating Watford 3-1 on Saturday, will visit either Blackburn Rovers or Coventry City.
Four fifth-round games ended in draws on Saturday while Arsenal and Cardiff play their fourth-round replay on Monday night.
It was an action-packed start at Goodison Park as Tim Cahill's header was handled on the line by Stilian Petrov only for the referee to play the advantage and allow 17-year-old Jack Rodwell to score in the fourth minute.
Villa levelled four minutes later with a James Milner penalty but a Mikel Arteta spot-kick restored Everton's lead by halftime and Cahill confirmed the end of Villa's 16-game domestic unbeaten run with a low shot 14 minutes from time.
Everton are in the quarter-finals for the first time in eight years and the first time in the career of David Moyes either as a player or manager: "It's been a long time coming but you don't get anything for reaching quarter-finals," the Everton boss told Setanta TV.
"We were struggling for players today, I was a bit worried, but they played tremendously well and earned their rewards.
United lost the away leg of their League Cup semi-final 1-0 to Derby last month before getting through 4-3 on aggregate but they were on top from the start on Sunday and got their reward when Nani cut in from the left before unleashing a fearsome drive in the 29th minute.
Everyone in the ground thought Cristiano Ronaldo had added a second six minutes later when he ran on to Ryan Giggs's flick off a long throw by goalkeeper Ben Foster but the referee eventually ruled the goal out for offside.
The second arrived when a Ronaldo free kick came off the wall to Darron Gibson, who steered in a low volley from 20 metres, and the Portugal winger got his goal three minutes after the restart with an excellent header from a corner.
A Miles Addison header gave the home fans something the cheer but Danny Welbeck swept home a well-taken fourth nine minutes from time.
"With the injuries we've had the squad has been getting us through those difficult games lately and it proved itself again today. I made some changes and they all did very well" manager Alex Ferguson told ITV.