Brezovan gifted Benjamin the only goal of the game on the stroke of half time after spilling a routine 20-yard shot and the veteran, who spent three months on loan at the County Ground last term, gleefully finished from four yards.
Swindon came into the game confident of recording their eighth win in nine games against a Peterborough team hammered 5-1 by Walsall in midweek.
The League Two leaders started on the front foot when the influential Paul Evans had a 20-yard shot pushed wide by visiting goalkeeper in Mark Tyler in only the third minute.
Adam Newton had Peterborough's first effort on goal when he volleyed wide in the sixth minute after Jude Stirling's enormous throw had only been half cleared by the Swindon defender.
Tyler was called into action again a minute later when he punched an inswinging corner over the bar, while the Posh shot-stopper was let off the hook in the 36th minute when his poor clearance fell at the feet of Christian Roberts but the Welshman dragged his shot wide.
Brezovan then came racing out of his goal to save at the feet of Newton in the 44th minute after he had raced onto Richard Butcher's centre, only for Benjamin to open the scoring soon after.
Roberts could have levelled 11 minutes after the restart when he seized on Stirling's poor back pass before thundering a right-footed shot against the inside of Tyler's left-hand post.
That was Swindon's last real chance and the pacy Newton could have secured the points 19 minutes from time when he brilliantly cut in from the right before shooting onto the roof of the net.