The opening half saw both sides create chances, the best when Kee was put clean through only to lift his shot over the crossbar.
That proved costly in the 32nd minute when Dean Winnard's loose back-pass saw Albert Adomah race clear, only to be chopped down by Dunbavin. The keeper was dismissed and John O'Flynn confidently dispatched the resulting penalty.
Barnet failed to make their numerical advantage tell and the visitors hauled themselves level after 59 minutes, Phil Edwards heading home Luke Joyce's corner.
The visitors had the perfect chance to take the lead when Kamdjo was sent off for bundling Kee to the floor, only for Jake Cole to save Edwards' spot-kick. The joy for Barnet was shortlived though when Kee struck his winner.