Horne's six penalties saw the Warriors move up to third place in the table.
The Blues desperately needed a good start to shrug off the memory of last weekend's record home defeat against Ulster, but Glasgow soon 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.
Glasgow were given another penalty following a long kick downfield by flanker John Barclay and from almost the same angle, Horne doubled the lead.
He repeated the trick three minutes later, with the Blues struggling to get into the opposition half.
The home side were being pinged repeatedly at the scrum and Italian referee Giuseppe Vivarini soon provided an opportunity for Horne to kick his fourth penalty.
Blues full-back Leigh Halfpenny, returning after a spell on the sidelines with an ankle injury, was able to put his team on the scoresheet with a penalty for offside.
Fijian prop Campese Ma'afu, who had struggled on his first start for the Blues, was replaced by Tongan Taufa'ao Filise at half-time as the Welsh region tried to shore up the set-piece.
The home side began to show some fire, with Josh Navidi prominent - but it was still Glasgow who scored next.
Vivarini ruled that the Blues had transgressed at the breakdown and the impeccable 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.
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