The visitors made a lively start and Clyde keeper Paul Jarvie was in action twice inside the first six minutes. Jim Lauchlan's fourth minute header was tipped over the far corner, and two minutes later a fierce effort from David Winters was pushed away to safety.
County kept up the pressure with further strikes from Gary McSwegan and Alex Burke, but it was Clyde who eventually broke the deadlock.
Tom Brighton made a strong surge down the left and Alex Williams, anticipating a near post delivery, left his marker standing to flick the ball past substitute goalkeeper Stuart Garden, who had replaced the injured Ian McCaldon after 13 minutes. County's response was a pile-driver from Mark McCulloch which Jarvie dealt with at the expense of a corner.
Lauchlan kept County in touch with an off the line clearance from a Brighton header as Clyde eased towards the break.
Williams found the net again two minutes after the restart but was clearly offside and later was unlucky to see a finish come back off the upright, but had he squared the ball in the first instance Brighton was well placed to score.
With a narrow lead Clyde were always vulnerable to a late equaliser and substitute Romauld Bouadji headed off the line in a crowded goal area.
In a match played in a good spirit, there were injury time bookings for County's David Cowan and Clyde scorer Williams with visiting substitute Stuart Taylor red-carded for a rash tackle on Eddy Malone.