Manager Ally McCoist had warned of complacency on their travels and his team responded impressively to secure their first away win of the season and rise above Elgin.
Dean Shiels opened up the account for the visitors (17) in stunning fashion as he curled in a cracker from the edge of the box.
But Gers withstood everything that was thrown at them and sealed the spoils with 10 minutes left when midfielder Lee McCulloch bundled the ball in.
The hosts threatened first when Stuart McColm left Chris Hegarty trailing in his wake before delivering a teasing ball into the box for Stefan McCluskey, but goalkeeper Neil Alexander punched to safety.
John Neill then sent a glancing header just past the post as Clyde passed up another chance to open the scoring.
Barclay was called into action again to block well at the near post from Andy Little, who then lashed the rebound across the face of goal and past the upright.
Rangers' pressure paid off as they took the lead when Shiels unleashed an unstoppable right-foot shot into the top corner.
The Bully Wee should have been back on level terms when Bryan Gilfillan's cutback found McCluskey but he somehow managed to fire off target.
If Barclay had come to the rescue of Clyde in the first half, Alexander impressed shortly after the restart with an excellent save to deny a point-blank header from McCluskey.
Clyde's hopes of a comeback were dealt a blow when they were reduced to 10 men after 68 minutes, Neill needlessly picking up a second yellow card for clashing with Shiels.
The game was over when Hegarty's cross found Ian Black at the back post with the lurking McCulloch to finish from close range.