Doncaster battled hard, playing with tremendous spirit and Billy Paynter hit the post, but they were unable to get the reward their efforts deserved in the League One clash at the Keepmoat.
The home side only had themselves to blame for conceding a couple of goals through slack marking at corners.
Right-back Jack King headed Preston in front in the 33rd minute after being left unmarked in the centre of goal following a corner from Joel Byrom before Rovers equalised from a free-kick on the stroke of half-time.
David Cotterill hammered home a swerving 20-yard shot that took a deflection off the head of Byrom.
Preston restored their lead in the 61st minute with Scott Laird heading home at the near post following another corner from Byrom and Beavon's clincher came against the run of play when he broke away to fire home from 20 yards with 18 minutes left.