The opening ten minutes belonged to Peterborough as Craig Mackail-Smith was twice denied by splendid saves from Stockport's Welsh goalkeeper Owain Fon Williams.
The hosts responded with Johnny Mullins heading narrowly wide from Tommy Rowe's deep corner, then defender Mullins was in the thick of the action again as his powerful shot was blocked.
It was the visitors who went in front on 22 minutes though, courtesy of an unfortunate own goal.
Aaron McLean crossed in low from the right and the ball was inadvertently directed past Fon Williams by Stockport skipper Michael Raynes.
It was Raynes who almost got his side back on level terms on 37 minutes when he rose highest to meet Anthony Pilkington's free-kick but his header went wide.
A minute before the break, Peterborough came close to a second when Fon Williams superbly saved a well-struck free-kick from Tom Williams.
Peterborough midfielder Shaun Batt had the first chance of the second period when he scooped over from just inside the Stockport box.
However, the hosts went on the level on 52 minutes. Rowe's corner wasn't cleared and there was Pilkington who lashed home the loose ball from 15 yards.
Ten minutes after that, Peterborough went back in front when George Boyd robbed Leon McSweeney on the left, before crossing for Mackail-Smith who volleyed home clinically.
The victory was wrapped up with 15 minutes left when Batt's cross was this time guided past his own keeper by young substitute James Tunnicliffe.