Sean Webb found space in the Dundee box in the opening minutes, but his low drive was held by Kelvin Jack.
Steven Craig's shot on the turn from six yards flew wide of the target after 15 minutes.
Iain Anderson saw his free-kick from 20 yards well saved by Stuart Garden with a quarter of the match gone.
Ross County came very close to opening the scoring 10 minutes before half-time when Steven McGarry crashed a close-range shot against the underside of the crossbar when he pounced upon a loose ball.
Five minutes into the second half Robbie Mann's header from Anderson's corner was saved by Stuart Garden. Midway through the second half Anderson made space for a shot at goal, but from 20 yards his right-foot effort went high over the bar.
A deep cross by Don Cowie was just out of John Rankin's reach when all alone inside the Dundee six-yard box with 15 minutes to go.
A head flick by Sean Higgins sent Steven McGarry through on goal with seven minutes remaining, and with a great chance to secure victory he fired his right-foot shot just past the post.