John Rankin had a shot saved by Bryn Halliwell and the Clyde keeper had to look smart again to turn an effort from Alexander Burke round the post after neat interplay between the striker and fellow font man, David Winters.
County took a deserved lead when Gary Bollan brought down Winters and Rankin fired a left footer into the corner of the net as Halliwell looked on.
Clyde eventually turned in a spell of pressure and, after Alex Burns had hit the side-netting, County keeper Colin Stewart turned a pile-driver from Craig Bryson away for a corner.
The resulting flag-kick produced a goalmouth scramble and Bryson slotted home the equaliser.
Clyde looked more lively after the restart and Jimmy Gibson had a shot stopped by Stewart before Burns wasted a free kick by shooting over the top.
After having less of the ball County pressed towards the end of the game and Halliwell saved a Rankin special at the expense of a corner with a minute left. Winters hit the bar after the corner and extra time followed.
In the first minute of overtime Clyde substitute Gary Arbuckle ran through the inside left channel to score from the edge of the penalty area.
Clyde failed to make sure of victory when Ian Harty missed from 12 yards out after he had been brought down by Martin Canning. However, Clyde hung on to the final whistle and now meet Celtic in the next round.