York goalkeeper Chris Porter got sent off after just 19 minutes after he hauled down Ian Craney just outside the penalty area when the Accrington striker was clean through.
From the resulting free-kick reserve keeper David Stockdale could only watch as Rory Prendergast's shot struck the woodwork and rebound to safety.
Paul Mullin scored the winning goal in the 55th minute when he volleyed home Steve Jagielka's teasing delivery.
Subs Not Used: Alcock, Howarth, Banim.