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.
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.
"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.
