A goal in each half from Nadir Ciftci, either side of Stuart Armstrong’s 48th-minute strike, put the visitors firmly in control.
County substitute Yoann Arquin pulled a goal back in the 76th minute, three minutes after coming on, but their hopes of snatching a point looked to be over when Liam Boyce was sent off with six minutes remaining.
However, Darren Maatsen set up a tense finale with a second goal for the hosts in the 89th minute, but United held firm in the closing stages to notch their fourth away win of the season and move within a point of leaders Celtic.
The defeat is the Staggies’ sixth in seven home matches this season and leaves them rooted to the bottom of the table.
County made a good start to the match and Joe Cardle flashed a shot across goal in the seventh minute but there were no takers.
At the other end United were awarded a free-kick in a dangerous position after Tony Dingwall brought down Conor Townsend but his effort flew over the crossbar.
The home side won a free-kick of their own in an equally dangerous spot but Martin Woods lashed his shot straight at United goalkeeper Radoslaw Cierzniak.
But it was the visitors who took the lead in the 21st minute when Chris Erskine sent Ciftci on his way and he evaded a challenge and rounded Mark Brown to fire into the empty net.
Armstrong almost made it 2-0 five minutes later but he was denied by the outstretched foot of Brown.
The visitors pushed for a second goal before the break and Ciftci almost netted again in the 40th minute but his header from a Ryan Dow corner kick was cleared off the goal-line by Steven Ross.
Woods tried to catch Cierzniak out with a long-range shot but the goalkeeper got down to his right-hand post to turn the ball behind for a corner.
But in a breakaway move in the 48th minute United went further ahead as Calum Butcher went on a driving run down the middle before laying the ball off for Armstrong who took the ball beyond Brown and passed it into the net.
County kept plugging away and a Jackson Irvine header in the box was turned behind for a corner by Jaroslaw Fojut.
United broke upfield in the 57th minute and Ciftci got behind the County defence but he shot tamely at Brown.
But the United defender made no mistake a minute later after Butcher put him clear, Ciftci shrugging off the attentions of Lewis Toshney and knocking the ball past Brown.
Maatsen made an instant impression after coming off the County bench when he raced up the right touchline and sent a high ball over to the far post which Cierzniak clutched out of the air.
But County pulled one back in the 76th minute when the ball fell for Arquin on the edge of the box and he beat Cierzniak low down to his left.
Arquin was almost in again a couple of minutes later when he raced in on goal but he lost control of the ball at the vital moment.
It got worse for County in the 84th minute when they were reduced to 10 men when substitute Boyce was shown a straight red card for a wild challenge on Paul Paton.
In a hectic finish County further reduced the deficit in the 89th minute when Woods slotted the ball through to Maatsen and he tucked a shot away from 12 yards.
In stoppage time United substitute Gary MacKay-Steven was denied by a point-blank save from Brown as the visitors clinched all three points.