Posh had Rovers' Enoch Showumni to thank as they claimed all three points at London Road, with the striker netting an own goal.
Showunmi contrived to steer a Grant McCann corner past his own Rovers keeper Tony Warner with 20 minutes remaining to complete a second-half turnaround.
Rovers were much the better side in the first period and were twice denied by home keeper Joe Lewis early on. He blocked from Showunmi and Lucas Atkins before the visitors were gifted the lead in the 24th minute.
Showunmi dispossessed the dithering Ryan Bennett and rolled the ball into the path of skipper John Welsh, who had no trouble slotting past Lewis from 15 yards.
That's the way it stayed until seven minutes after the interval when Grant McCann hauled Posh level. The Northern Ireland international finished smartly with a first-time effort from 15 yards after George Boyd had guided a Lee Tomlin pass into his path before Showunmi's own goal moved Posh up to fourth.