Glasgow had been forced into a pre-match change when Jack Mann was withdrawn from the line-up, with Henco Venter stepping in at number eight.
Venter was involved as Warriors moved in front after just five minutes, one of several willing carriers who took the ball close to the line before Nathan McBeth finally crossed.
The windy conditions made it difficult for both sides but Glasgow were clearly coping better and soon landed their second try.
Horne was the creator with quick ball from the ruck, setting up Jamie Dobie, who dived over from close range.
Dobie thought he had scored again after taking a whipped pass from Adam Hastings and diving over in the corner but the Lions had men across to defend and the TMO’s decision was no try.
Dobie would not be denied for long, however, as Warriors underlined their dominance with a third score.
Hastings played the pass but the rest was down to the Glasgow wing who stepped inside a gap, fended off two attempted tackles and sped over the line.
It was one-way traffic at this point and it was Horne who landed the bonus point with Glasgow’s fourth try after taking a pass from Dobie, making history in the process.
Warriors were soon celebrating a fifth try, Hastings throwing a dummy then easing through a gap to score.
Lions were then reduced to 14 men for 10 minutes when scrum-half Nico Steyn was shown a yellow card for a team offence and Glasgow took advantage when Grant Stewart burrowed over from close range.
The second half was a scoreless affair but the damage had all been done in the first half with Glasgow eight points behind Leinster and four clear of the Bulls in the race for play-off places.
