The visitors started brightly and after 10 minutes Ryan Lowe and Tommy Rowe combined but the latter's weak shot from close range was tipped wide by Barnet stopper Lee Harrison.
Stockport continued to press and with 22 minutes on the clock Rowe found Stephen Gleeson in space only for a brilliant saving tackle from Josh Wright to deny the on-loan Wolves man.
The deadlock was broken within two minutes of the restart as Jason Puncheon drilled the ball home from 20 yards after his initial free-kick had been blocked by the Stockport wall.
But County were not behind for long and in the 51st minute the lively Rowe capitalised on an error from Barnet defender Michael Leary to coolly lob the advancing Harrison from the edge of the box.
And that was the way it stayed until the 81st minute when sub Danny Hart crossed from the right and Akurang found the net with a bullet header that gave John Ruddy no chance.
Prior to the goal Gleeson had tested Harrison while Adam Birchall had forced Ruddy into action moments before Akurang's winner.