Jeff Hughes put County ahead from the penalty spot at Hillsborough, but second-half strikes from Julian Bennett and Gary Madine turned the League one game on its head.
Owls midfielder Chris Lines needlessly handled Mike Edwards' searching free-kick in the 10th minute and Hughes stepped up to slot home his second penalty in successive games.
Wednesday levelled soon after the restart when County failed to clear Lines' corner and Clinton Morrison teed up left-back Bennett to volley home.
Madine then put Wednesday ahead in the 67th minute with a close-range finish into the roof of the net following Buxton's low cross from the right.
Magpies substitute Craig Westcarr wasted a golden opportunity to equalise 15 minutes from time before skipper Neal Bishop received his marching orders two minutes later for a second yellow card following a foul on Jose Semedo.