The well-travelled hitman pounced 53 seconds into the second half, firing into the root of the net from close range.
County had their fair share of chances, but Mark Tyler had an excellent game in the Peterborough goal and produced a string of smart stops to deny the hosts.
Posh started brightly and came close to taking an early lead when Benjamin's left-foot strike from the edge of the box was well saved by John Ruddy between Stockport posts.
At the other end, and with their first attack of the game, striker Adam Proudlock nearly netted on his first League start for the Hatters.
Winger Michael Malcolm whipped in a delightful cross from the left edge of the area, and Proudlock was unlucky to see his acrobatic bicycle-kick fly over the crossbar.
County had another good chance, this time in the 17th minute, after both skipper Keith Briggs and loan signing Glenn Murray had shots charged down from close range.
Eventually the ball ran kindly for Jason Taylor, but the midfielder's powerful right-foot shot missed the target by the tightest of margins.
Peterborough pressured and Benjamin forced Ruddy to make another excellent save. The big centre-forward turned right-back Rob Clare and unleashed a fierce left-foot drive, only for Ruddy to stand tall and avert the danger.
Taylor should have done better with another good chance after 37 minutes, when he dragged a right-foot shot wide of the target despite having plenty of time and space.
Two minutes later Stockport were denied by a brilliant piece of goalkeeping. Murray collected the ball from Malcolm and rode a tackle before seeing his toe-poked shot from six yards out blocked by Tyler.
Some sloppy defending at the start of the second half saw Benjamin give Peterborough the lead, before Briggs nearly equalised two minutes later with a rasping drive.
The impressive Richard Butcher teed up Danny Crow in the 52nd minute, but Ruddy made a brilliant point-blank block.
Crow saw another shot blocked from close-range soon afterwards, before Briggs fired a left-foot shot wide with 20 minutes remaining.
But it was Benjamin's goal which proved the difference as Posh took the points.