The home side produced a storming second-half display after coming back from a goal down to help push themselves further away from the relegation mire.
And while one run ended with Stockport's defeat, another continued as Barnet's Oliver Allen – son of former Spurs legend Clive Allen – scored for the fourth-game running to bring Barnet level late in the first-half. Barnet's Dean Sinclair and Liam Hatch then found the net after the break.
But it was the rampaging visitors who made the brighter start. Midfielder Dominic Blizzard forced the first save of the match after eight minutes when his drive was tipped over by Ross Flitney in the Barnet goal.
County finally got the goal they deserved shortly after the half-hour mark.
Striker Tes Bramble, who had recovered from a virus in time to play, beat Flitney at the near post to head home Michael Rose's in-swinging corner from the right for his seventh goal of the season.
That should have galvanised Stockport into taking full control but the visitors took their foot off the pedal, allowing Barnet to level.
Allen got the goal with a cracking strike from the edge of the box a minute before the break, showing that goal scoring runs in the family.
And it was the first goal keeper Wayne Hennessey had ever conceded in his league career and the first Stockport had let in since Watford's Moses Ashikodi found the net on January 6.
It got even better for Barnet two minutes after the beak when midfielder Dean Sinclair sent a rasping left-footed drive, which took a slight deflection, into the bottom corner to give the hosts the lead after Stockport had failed to clear.
Barnet's win was sealed when Hatch ran clear to fire home with eight minutes left.