The result might have been rather different if County have taken an early chance through full-back Gary Irvine, but full credit to keeper Peter Cherrie for the intial block and defender Craig McKeown for sweeping the re-bound to safety.
A dreadful clearing kick by County keeper Craig Samson was very nearly punished by Eddie Malone's left-foot shot from 30 yards. Samson managed to redeem himself in the 36th minute when he produced an amazing reflex save to push away Neil McGregor's acrobatic volley.
As half-time approached, Alex Williams tested Cherrie from long range and Clyde mid-fielder Paul McHale was denied by the post after beating three defenders at the end of a dazzling run.
McGregor fluffed a free header as Clyde made a fine start to the second period, and the opening goal soon followed as Andy Dowie handled a deflected shot from Stephen O'Donnell and McHale converted the penalty kick.
County should have equalised before the hour mark when Cherrie could only parry a fierce drive by Michael Gardyne into the path of Martin Scott, but the substitute blazed the ball over the bar.
Andy Ferguson's blistering twenty-five yard rocket then made it 2-0 and fellow striker Dougie Imrie completed the scoring five minutes from time with a scrambling header at the back post. .