Clyde had all the pressure in the opening stages and seemed to be a yard quicker to the ball than their opponents in all areas of the park.
The home side came close on a couple of occasions before Eddie Malone began the scoring after 18 minutes.
Kevin McDonald hammered a free-kick off the Brechin defensive wall and the ball bounced kindly into the path of Malone, who found the top corner of the net with a shot from the edge of the area.
Brechin's determination brought them an equaliser after 35 minutes. Clyde elected to take a free-kick although injured player Robert Hall was still down.
However the move backfired and Clyde had the cheek to protest after Chris Templeman's cross was turned over the line by Stuart Callaghan to level the match at the break.
After the restart Williams' quickfire treble ended the game as a contest. The first goal came from a clever pass into the box by Malone and the second from a header from a cross supplied by Dougie Imrie.
Williams' third followed a lovely through ball from Craig Bryson, giving him ample time to tuck the ball past the stranded Craig Nelson.
The rout was completed followed a slick move between Stephen O'Donnell, Tom Brighton and Bryson, with O'Donnell providing the clinical finish.