County got off to a flyer with a goal from John Rankin after just two minutes. Rankin, who scored his first goal at Victoria Park last week, curled the ball high into the net from 20 yards for a wonder strike.
Just before that Alex Burke forced a fine save from Clyde keeper Paul Jarvie and David Winters had a header cleared off the line by Robert Harris.
Before the Bully Wee could draw breath, County doubled their lead. Lionel Djebi-Zadi crossed from the left and Stuart Taylor's knock down was scooped high into the net by Burke.
But Clyde pushed more players forward and the tactic paid off in the 31st minute when a hopeful ball into the County box caught the home defence flat footed and Craig Bryson nipped in to send a low drive into the corner of the net.
County made an early change at the start of the second half with Sean Higgins coming on for Winters.
The game came to life again on the hour when County keeper Ian McCaldon pulled off a double save from Tom Brighton and Alex Williams, and at the other end Burke was clean through on goal but Jarvie blocked his effort on the edge of the area.
A goal looked on the cards for the home side and it came in the 67th minute when Rankin found Steve McGarry on the right and his ball into the box was bundled over the goal line from close in by County substitute Sean Higgins.