The victory booked a fourth round tie at home to fellow top flight side Fulham.
Rooney scored 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.