Clyde got off to a flyer when former Ross County player Stephen O'Donnell shot high into the net over Stuart Garden in the seventh minute from just outside the area.
The County defence were slow to clear their lines and Alex Williams set the ball up nicely for O'Donnell's finish.
The County defence were at sixes and sevens a couple of minutes later when John Rankin made a goal-line clearance from a Tom Brighton shot.
O'Donnell was looking up for it and he forced Garden to claw the ball behind for a corner when he tried to curl one in at the far post.
Then Alex Williams was clean through on Garden, but the goalkeeper did well to force the Clyde striker wide and Chris Higgins blasted over from a cut-back.
County were finding it difficult to get into their opponents half as the Bully Wee dictated the early part of the game.
County were committing basic errors at the back. When Mark McCulloch gave the ball away in the middle of the park, Craig Bryson latched on to it and his lay-off for Brighton almost resulted in a second goal, but his shot was cleared.
County had their first effort on target in the 38th minute when Rankin set Lionel Djebi-Zadi free on the left and the Frenchman's low cross was met by Alex Burke, but Clyde keeper Peter Cherrie turned the ball behind for County's first corner of the game.
In County's first attack of the second period, Burke raced through on goal but he was quickly closed down by Cherrie, then Gary McSwegan had a go only for his effort to be blocked.
But the home side were a poor second best to Clyde who looked content to safeguard their slender lead.
Stuart Taylor, who had earlier replaced Fergus Tiernan in the County midfield had a chance to level the scores in the 80th minute when he spotted Cherrie off his line, but his attempted lob from well out on the right bobbled inches past the far post.