Everton were highly impressive as they claimed a 3-1 victory over Villa and while David Moyes has played down talk of cup glory, a home tie in the quarter-final will surely encourage the Scot.
United always had too much for Derby at Pride Park and a 4-1 victory will only boost growing confidence at Old Trafford, as Sir Alex Ferguson's side continue to chase five trophies.
A day of stalemates on Saturday saw only Chelsea book safe passage to the competition's last eight, via a 3-1 win at Watford, and Guus Hiddink's side will now play away from Stamford Bridge against either Blackburn or Coventry, who drew 2-2.
The Sky Blues were just moments away from winning at Ewood Park but Christopher Samba's injury-time goal saw Rovers claim a 2-2 draw.
Chelsea beat Blackburn in the semi-final en route to their cup triumph two years ago with Michael Ballack scoring the winner in extra-time.
Arsenal play a fourth round replay against Cardiff City at Emirates Stadium on Monday, and should they win through Burnley await in the fifth round.
The winner of that tie will play either Sheffield United or Hull City, who could not be separated at Bramall Lane on Saturday, at the quarter-final stage.
FA Cup quarter-final draw:
Blackburn or Coventry v Chelsea
Swansea or Fulham v Manchester United
Cardiff/Arsenal or Burnley v Sheffield United or Hull City
Everton v West Ham or Middlesbrough
Ties to be played on the weekend of March 7-8.