The home side began well at the Global Energy Stadium, putting high-flying United under some early pressure as they looked to move themselves off the bottom of the table.
But the visitors were ruthless on the counter-attack and Nadir Ciftci eventually put them ahead in the 21st minute, steering home after rounding goalkeeper Mark Brown.
The lead was doubled three minutes after the interval, as Stuart Armstrong latched on to Calum Butcher's delightful pass before stroking in, dealing County a cruel blow so early in the second half.
Butcher played a major role again 10 minutes later as he set Ciftci up for his second of the day, with United seemingly cruising to an easy victory.
Yoann Arquin pulled one back with just under quarter of an hour to go and, although substitute Liam Boyce earned a straight red card for a foul on Paul Paton soon after, Darren Maatsen struck a minute from time to ensure a nervous finish for United.
Nevertheless, they held on to the victory and ultimately climbed above Inverness Caledonian Thistle into second, one point behind Celtic, who travel to Motherwell on Saturday.
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