Although Clyde completely dominated the first 45 minutes, they seldom threatened the visitors and were just one goal ahead at half-time from a scrappy effort.
Eddie Malone took a corner from the right side which resulted in an untidy goalmouth melee which ended with Alex Williams forcing the ball over the line at the far post with Brechin defenders protesting. The linesman was in no doubt that the ball was over the line and the effort stood.
Clyde might have had a second four minutes from the break when Tom Brighton set on a mazy run past five defenders, but the finish was straight at keeper Craig Nelson.
Clyde received a rude awakening four minutes after the restart when Kevin Byers equalised with a 30-yard rocket.
Steven Hampshire began the move with a long ball out of defence towards the head of Gerry Britton. The front man missed his jump, and although he didn't touch the ball he deceived his defender, which gave Kevin Byers his chance.
The visitors' joy was short lived as Clyde surged back and ended a a two-minute spell of pressure on the Brechin goal by taking the lead.
Britton provided the cross and Michael McGowan's sweet left-foot finish turned out to be the winning goal.
It was probably fitting that McGowan should grab the winner as he was the best player on view all afternoon.