The doomed home side have now lost eight games on the spin after another miserable afternoon.
With Rovers still eyeing the slim possibility of making it into the play-offs, they got off to a perfect start with a goal after just 63 seconds.
Michael McIndoe lofted in a deep cross to the far post, and little Dave Mulligan nodded the ball invitingly into Ricky Ravenhill's path. All the midfielder had to do was poke the ball past County goalkeeper Neil Cutler from all of eight yards.
Soon after Mulligan tried his luck from distance, but this time Cutler was equal to the task.
Young Stockport striker Adam Le Fondre glanced a header wide at the other end, and with Rovers seemingly content to play keep ball in and around the midfield, it was the relegated side who had the next decent chance.
Ravenhill failed to clear a Danny Jackman cross into the box, but Ian Turner got down well to block Kevan Hurst's low drive.
However Stockport did get themselves a goal on 28 minutes when Keith Briggs struck a shot from 25 yards and the ball was inadvertently diverted into the goal by Damien Allen for the equaliser.
Cutler was acrobatic when denying Leo Fortune-West with a brilliant one-handed save on 34 minutes, but after Chris Turner's side has twice gone close through veteran striker Stuart Barlow, they did grab a second goal on the stroke of half-time.
Hurst's cross was flicked on by Barlow, and there was teenager Le Fondre who headed home clinically at the near post.
Amazingly Stockport should have gone two clear three minutes after the restart when Ravenhill beat out Barlow's near-post header near enough on the line.
Paul Green was having an effective game in the Rovers engine room, and on 61 minutes he went agonisingly close to a goal when his low drive bounced back off the base of a post.
Rovers finally began to smell the possibility of maximum points, and late strikes from Green, McIndoe and young debutant Adam Brown completed what in the end was a comfortable victory.