The visitors found themselves 2-0 down inside half-an-hour, but fought back to take all three points in what turned out to be a pulsating encounter. But they had themselves to blame for being behind in the first place.
Tresor Kandol, who created havoc for the Magpies' defence, won a penalty on 25 minutes when keeper Kevin Pilkington charged out and brought him down in the box.
Captain Ian Hendon converted against his old club adding to the two spot-kicks he scored against them last season.
And Barnet doubled the lead on the half-hour when Pilkington crashed into his own defender Alan White to spill the ball into the path of Jason Puncheon who pounced to tap in.
That seemed to spur the visitors on and White made amends two minutes later heading home Andy Parkinson's corner from close range. And within five minutes, they were level.
White's long throw was flicked on by Lee and Parkinson volleyed into the bottom left-hand corner from 18-yards.
County had the chance to take the lead minutes later when Hendon was judged to have brought down Lawrie Dudfield in the box. But justice was done when Lee Harrison dived to his left to save Dan Martin's penalty.
Magno Vieira had a glorious opportunity to take the lead for Barnet seconds after the re-start, but put his shot wide after being put through one-on-one with Pilkington.
Kandol had the ball in the net on 48 minutes from Simon King's cross, but was ruled out for offside.
But it was Lee who was celebrating seven minutes later after beating the offside trap from David Pipe's through ball to slot past Harrison.
Junior Mendes nearly made it four on 67 minutes firing high, but Harrison turned the ball over well for a corner.
Three minutes from time, substitute Giuliano Grazioli had the chance to equalise for Barnet, but he put his shot wide.
County fought off late pressure from the home side, and held on for the three points.