The in-form visitors claimed the first sight of goal when Jason Norville produced a neat turn but his shot flew over the crossbar.
At the other end, Anthony Elding crossed for strike partner Matty McNeil, only for his shot to be deflected wide for a corner.
On 13 minutes Barnet's Jason Puncheon produced a wild shot which flew well over the crossbar, but within a minute he curled in a great low cross which Adam Birchall just reached as he poked the ball past Conrad Logan for the opening goal.
Stockport tried to respond quickly and debutant Leslie Thompson missed a great chance from just inside the box, but the home side did level matters on 25 minutes.
Thompson streaked away down the right and was clumsily tripped close to the byline by Sagi Burton.
Referee Haywood pointed to the penalty spot and Elding made no mistake for his seventh goal of the season.
On 36 minutes Stockport went close to a second goal when Elding rose well to meet Adam Griffin's out-swinging cross, but his header was deflected over by centre-back Burton.
Barnet's Josh Wright shot well over just before the break.
Within two minutes of the restart, Barnet had gone back in front. Again Puncheon crossed low from the left, Logan failed to gather the ball and there was Birchall who rammed home his second goal from 10 yards.
On 55 minutes the Bees again went close when Norville's half-volley flashed just past the far post, but they did make it 3-1 on 62 minutes when Nicky Nicolau's crossed was finished off by Wright at the far post.
Carew added a fourth, then Liam Dickinson's stoppage-time strike was a mere consolation for the hosts.