The early goal seemed to give United extra impetus and they took control of the game with Antonio Valencia a regular threat down the right, with Rooney stinging the palms of Given from nine yards out.
A second goal for the Red Devils looked on the cards and they almost got it when Valencia fired in a low cross that Welbeck diverted towards goal; Given was however equal to the effort as he reacted quickly to flick over.
Welbeck did find a way past Given two minutes before the interval by getting on the end of Patrice Evra's low cross and prodding home to make it two.
After the break, United struggled to dominate in the same manner as the opening half and the first real chance fell to Villa substitute Emile Heskey, with his header being tipped over by De Gea.
Midway through the second 45 Valencia skimmed a low ball to Rooney in the area but the England forward, wasteful for much of the game, dragged his effort wide.
He made up for it minutes later by exchanging passes with Valencia before seeing his low drive deflect past Given to wrap up the points. Rooney was not offered the chance to complete his treble as he was instantly replaced by Dimitar Berbatov.
Substitute Nani added a fourth in stoppage time by slotting under Given following a pass from Jonny Evans.
United next welcome Everton to Old Trafford, while a loss for Villa means they can still possibly be dragged into a relegation battle at the bottom.