After a goalless first half at the Global Energy Stadium, Iain Vigurs gave County the lead in the 48th minute with a glancing header.
Johnny Russell equalised within four minutes after he beat County goalkeeper Michael Fraser to a long ball on the edge of the 18-yard box - and United grabbed the winner in the 65th minute when Stuart Armstrong nodded in a Jon Daly knockdown.
United made a positive start to the match and after a Barry Douglas free-kick was only partially cleared, the ball fell for Russell in the box but his snapshot was blocked by County goalkeeper Michael Fraser.
The home side then pressed and when Gavin Gunning lost possession on the edge of the 18-yard box, Rocco Quinn forced a fine save from Radoslaw Cierzniak.
United won a free-kick in a dangerous position in the 25th minute and Fraser had to be alert to tip Douglas' attempt over the crossbar.
The visitors were seeing more of the ball but it was County who almost took the lead in the 28th minute. Vigurs picked out Stuart Kettlewell with a defence-splitting pass but his shot was cleared off the line by Brian McLean.
The chances kept coming and when Tokely played Sam Morrow in, the big County forward looked sure to score but Cierzniak pulled off a fantastic point-blank save.
In the last action of the first half, a slack backpass from Gunning almost let Vigurs in but Cierzniak was quick off his line to clear the danger.
County broke the deadlock in the 48th minute when Vigurs scored with an acrobatic header after he got on the end of a superb cross from Richard Brittain.
But it was all-square four minutes later when Russell raced on to a long ball and slid the ball past the advancing Fraser into the corner of the net from the edge of the 18-yard box.
The goals kept coming and United got their noses in front in the 65th minute when a deep cross from Gary Mackay-Steven was knocked down at the far post by Daly and Armstrong stooped to head home from close range.
County rang the changes up front with Colin McMenamin and Steve Ross coming on but it was Vigurs who came close to grabbing a goal back with a long-range shot in the 77th minute.
Right at the end of this contest, Skacel was denied at point blank range by Fraser, after substitute Michael Gardyne played him in.