Paul Taylor flashed a shot just wide as Posh started brightly, although Daniel Johnson saw an effort deflected over at the other end.
Andreas Weimann fired wide as Villa continued to press, but visiting keeper Shay Given was tested by a fierce drive from Taylor.
The visitors broke the deadlock just before the break when Weimann's cross was deflected into the path of Johnson and the young midfielder tapped home.
Villa's second goal arrived 10 minutes from time when substitute Brett Holman finished off a patient move.
Holman was denied a second goal in the closing stages by a superb save from United keeper Joe Day, who had replaced Bobby Olejnik at half-time.