The Blues only response was a penalty by Leigh Halfpenny just before the break but the die had already been cast by then.
The Blues were bitterly disappointing in their last game, a record home defeat against Ulster, and were looking to start with a bang but the Scottish side quickly established control up front and were rewarded with points on the board.
Right-wing Tommy Seymour was dragged down by opposite number Tom James after breaking the first line of defence, but the visitors maintained their pressure and were awarded a penalty which Horne slotted over from wide out. Poor discipline by the Blues gave Horne two more opportunites to try his luck at the posts and he was successful on each occasion to stretch Glasgow’s lead.
Italian referee Giuseppe Vivarini didn’t impress the home fans with a constant stream of penalties at scrum time, with Horne banging over his fourth penalty. Halfpenny eventually got Phil Davies’ side on the board with a successful penalty.
The second half didn’t start any better for the home side when Horne kicked a 45-yard penalty.
The home fans were happier when the official decided that it was time Glasgow took the blame for a disintegrating scrum, but Halfpenny’s long-distance kick slid wide. And it was soon back to the familiar pattern as the Warriors destroyed the home pack to give Horne another three points for his collection.
The Blues came close to the game’s first try when Filise burst up the middle and the ball was spread wide to Halfpenny, who raced for the corner. But just as the Wales international was diving for the line, the covering Stuart Hogg managed to knock the ball from his hands.
Glasgow were reduced to 14 men when hooker Dougie Hall was yellow-carded for a professional foul, while Horne proved that he was only human when he drove a straightforward penalty wide of the posts. Not that it mattered. Glasgow kept the Blues penned in their own half until the final whistle heralded a solid victory.