County started strongly and a John Rankin free-kick in the fourth minute caused the Dundee defence problems, but Scott Murray managed to fend off Sean Higgins as he challenged to win the ball on the goal-line.
Dundee tried to get into the game and Steven Robb ran into the opposition box, but he over-ran the ball and County keeper Ian McCaldon gathered.
Then when Steven Craig was presented with a chance for the visitors in the 14th minute, he failed miserably to hit the target and his shot ran out for a throw-in.
But it was mostly County and Stuart Anderson, making his home debut, got a shot on target after Rankin laid the ball on for him.
The dark blues were getting closer to a goal and after David Cowan made a mess of a clearance, Craig missed a great to score when he shot straight at McCaldon from 12 yards.
The chances were mounting up in front of the County goal and centre-back Callum MacDonald moved upfield to fire a shot wide of McCaldon's left-hand post.
County mounted the first attack of the first half, but Higgins blasted his shot wide of Murray's left-hand post.
Higgins was out of luck again when his goalbound shot was charged down by Bobby Mann and at the other end Bryan Deasley was blocked.
The game was dying a death when Rankin tried a long-range effort that Murray did well to get down to.
County made a double substitution in the 73rd minute which saw the introduction of David Winters and Craig Gunn.
Winters had a great chance to break the deadlock six minutes later when the ball fell nicely to him after Mann failed to cut out a Lionel Djebi-Zadi cross, but the County substitute shot weakly into the arms of Murray.
Winters was looking lively and he was not far away when he cut from the left and fired a shot just wide of the far post.
County forced a free-kick in the final minute, but Rankin's effort flew wide of the upright.