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.
