The boot of full-back Isa Nacewa extended that lead to 19-13 before a Duncan Weir penalty and Horne's try secured an improbable victory to get the Warriors' season up and running.
The feel-good factor from Ireland's victory over Australia in the World Cup looked like it was going to roll over at the RDS with Nacewa scoring two penalties while Fionn Carr and Luke Fitzgerald made two promising breaks.
Scott Wight, having seen an earlier attempt come back off the post, got the Warriors on the board with a penalty in the 22nd minute.
Eight minutes later Glasgow took the lead for the first time as a scrum on the Leinster 22 resulted in lock Nick Campbell found the space to barge his way over the line after some good carries by the Warriors' front row.
That was part of a good spell the Warriors were enjoying but just five minutes before the break a nice backline move resulted in Dominic Ryan offloading for Ian Madigan to touch down with Nacewa converting.
Weir replaced Wight at the interval and immediately got the Warriors on level terms with a penalty only for Nacewa to reply in kind.
Ryan was then carried off with a serious-looking ankle injury before Glasgow captain Ryan Harley, who had a fine game, blotted his copybook by being sent to the sin-bin.
Nacewa and Weir exchanged further kicks in his absence before the Warriors caught the hosts with a sucker punch.
Leinster had been knocking at the door but failing to find a way through when Glasgow broke downfield through Stuart Hogg who fed Horne who had only on the pitch for a matter of minutes, to dive over with Weir converting.
With 13 minutes remaining, Leinster threw everything at Glasgow but their defence held firm.
However Joe Schmidt will be left to rue his side's inability to turn possession and territory into points earlier in the match as Glasgow's tenacity proved decisive.
|Leinster Rugby Score Card|
|Glasgow Warriors Score Card|