Birmingham took the lead after 17 minutes, Ricardo Swaby took a right wing corner, which was deflected into his own net by James Wild, who then levelled for his team when he prodded home a knock back from Sam Golden after 25 minutes.
Three minutes later another Ricardo Swaby corner was cleared out to Danny Brain who smashed the ball home from 25 yards.
Birmingham sealed the points after 52 minutes when Simeon Cobourne seized on a loose ball and crossed for Saul Hodgetts to score unmarked from 6 yards.