Victory saw Premier League leaders United, the record 11-times FA Cup winners, into a fourth round tie at home to fellow top flight side Fulham.
Rooney opened the scoring in the ninth minute after the recalled Anderson produced a superb, defence-splitting pass and Javier Hernandez bore down on goal before unselfishly squaring the ball to the England striker, who was never going to miss from a few yards out.
The forward could have put the result in this all-Premier League clash beyond doubt in the second half but Rooney blasted over the top from the penalty spot after Jordan Spence handled a cross from Ryan Giggs.
Earlier, Arsenal left it late before seeing off Swansea 1-0 in a third round replay with England midfielder Jack Wilshere scoring four minutes before full-time with a powerful shot. “It was a great set-up from Olivier Giroud,” said Wilshere. His goal earned Arsenal —bidding for a first major trophy in eight years -- a fourth round clash with second-tier Brighton, the 1983 FA Cup finalists.