Iain Davidson gave Dundee the lead eight minutes from the break, but County hit back strongly and substitute Gary Glen fired home a deserved equaliser with 16 minutes remaining.
Both sides took time to settle and it took until the 15th minute for the first real effort on goal. County's Alex Cooper got behind the Dundee defence and his cross into the box was knocked back by Colin McMenamin for Mark Fotheringham, whose low drive flew wide of the upright.
At the other end a deep cross from Matt Lockwood in the 20th minute fell for Nicky Riley at the far post, but he failed to control the ball and the chance was lost.
County charged upfield and a mix-up in the Dundee defence allowed Cooper to have a shot at goal, but the ball deflected off Irvine and behind for a corner.
But with their first shot on target Dundee took the lead in the 37th minute. The ball broke to Davidson 25 yards out and his shot left County goalkeeper Mark Brown a virtual spectator as the ball flew high into the net.
The goal galvanised the visitors and they had a chance to double their advantage when Jim McAlister fed the ball out to Riley, but his shot flashed into the side-netting.
County came close to an equaliser four minutes from the break when Ross Tokely met an Iain Vigurs corner at the back post but his point-blank header produced an instinctive save from Rab Douglas.
County made a lightning start to the second half and after just 15 seconds Vigurs fired a deep cross-cum-shot which Douglas was forced to touch over the crossbar.
Vigurs tried a long-range shot in the 49th minute but Douglas got down well to hold on to the ball. The same player almost sneaked one in at the far post when he delivered another deep cross a minute later, but the ball just cleared the far post.
Dundee were not out of it, however, and John Baird invited a late challenge by Fotheringham who found himself in referee Alan Muir's book.
County got back into the match in the 74th minute after a sustained spell of pressure from the home side. A Kettlewell shot on the edge of the area fell nicely for Glen and the substitute flashed the ball high into the net from the edge of the six-yard box for a deserved equaliser.