Steven Saunders and Ivan Sproule both saw red for the visitors, who trailed to strikes from Kenny McLean and Conor Newton either side of half-time.
Saunders reduced the deficit but followed Sproule down the tunnel after two quick-fire yellow cards, ruining County's chances of a comeback.
The sides had been separated by just a point going into the clash, which was rearranged after the original tie was abandoned midway through the second half on November 2 due to a waterlogged pitch.
McLean put his side in front after 34 minutes, beating goalkeeper Mark Brown from the edge of the area.
Newton extended the hosts' lead six minutes into the second half, Newton collecting Steven Thompson's pass before rolling past Brown.
Hope turned to despair two minutes later, however, as Sproule was shown a straight red card for a dangerous high challenge on Marc McAusland.
Saunders was then handed two cautions in the space of three minutes to leave County with nine men before going down to a third straight defeat.
