The visitors drew first blood on six minutes when Claudiu Tipote clipped the ball past the keeper and followed in to smash the ball home.
But within four minutes United had levelled when Sam Holmes rose to head home an excellent cross.
The goals kept coming and it was Edgware who went ahead on 17 minutes when Mark Burgess powered home a header at the far post from a corner.
This time United had to wait longer to strike but managed to do just that midway through the second period. Town were penalised for a sliding challenge in the box and Tom William stepped up to fire in the penalty low into the corner.
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United sit tenth in the table, on the same number of points as Edgware but having played four fewer games.