A Sean Higgins header was all that separated the sides, although County thoroughly deserved the three points.
County won a free-kick in the first minute but Derek Adams' floater into the box was cleared for a throw-in.
The home side were enjoying most of the play in the early stages and a Higgins cut-back in the seventh minute was blasted over the crossbar by Kevin McKinlay.
At the other end, Robert Campbell broke through the County defence but he played the ball too far ahead of him and Craig Samson gathered.
Danny Griffin limped off in the 18th minute and was replaced by Callum MacDonald.
County came close again four minutes later when they won their third corner of the game. Adams swung the ball over and Mark McCulloch's header was cleared off the goal-line by Paul Dixon.
Don Cowie should have given the home side the lead in the 35th minute when he was put clear on goal by Michael Gardyne but he placed the ball wide of Ludovic Roy's right-hand post.
But Cowie made up for his miss a couple of minutes later when he supplied the final ball for Higgins to glance a header wide of Roy.
Cowie had another chance to score early in the second half but he shot straight at Roy, then some head tennis in the County box ended with an Andy McLaren shot that was blocked.
Dundee were seeing more of the ball but County broke in the 55th minute and Gary Irvine drifted a ball just over Roy's far post. In the next attack, Higgins rounded Roy but the County striker shot into the side-netting.
Dundee were awarded a free-kick just outside the area midway through the second half and Gavin Swankie was not far away with a shot that flew inches over the crossbar.
But it was mostly County and Gardyne tested Roy with a shot that the keeper managed to scramble behind for a corner.
Adams sliced the final chance of the game into the side-netting as County finished on top.