Craig Bryson burst into the County box in the second minute but his shot at goal was blocked by Martin Canning.
After a poor start to the game County almost made the breakthrough in the 21st minute when Cowie and Sean Kilgannon linked up well on the left but the final ball into the box fell behind the inrushing McGarry and the chance was lost.
But a terrible mix up in the County defence cost the home side dear. Neil MacDonald lost his footing 15 yards out and Bryson nipped in to steer the ball past Stewart.
Stewart who had taken a couple of knocks in the game was forced to retire in the 27th minute with Stuart Garden taking over in goal.
McGarry came close when he got on the end of a Kilgannon cross in the 34th minute but his flashing header flew just past the upright.
Then Halliwell denied McKinlay at point blank range when he scooped away the youngster's goalbound header.
Allan Morrison replaced Bryn Halliwell in the Clyde goal at the start of the second half, and Adam Moffatt came on for Kilgannon in the County line-up.
Clyde looked content to hit on the break and Alex Burke was introduced in the 58th minute as County went all for an equaliser.
Burke was involved in most of County's best moves since coming on and it was the former Queen of the South player who levelled in the 69th in spectacular fashion.
Cowie got behind the Clyde defence and his inch perfect cross was headed past stand in keeper Morrison at the far post.
County continued to press and Rankin fired a long range shot over the top, then Martin Canning headed agonisingly wide when he climbed to connect with a Cowie corner.