After withstanding a strong opening from the hosts Leinster came closest to opening the scores when Gary Brown went close twice and it was only desperate defence that kept the scores level.
The Irishmen had the momentum though and flyhalf David Holwell eventually broke the deadlock with a close range penalty.
Glasgow rallied strongly and Andy Henderson and Kevin Tkachuk both came close to grabbing the opening try of the game.
Their side's endeavours did bring the scores level soon after when Dan Parks slotted an easy penalty when the Leinster back-row were penalised at a scrum.
Leinster took control of the game before the break with a try from Girvan Dempsey.
The Dubliners pounced on a sloppy clearance from Calvin Howarth and some good work from Brown and Gordon D'Arcy created the opportunity for the Irish international to cross for the five-pointer.
Holwell added the conversion and also an injury-time penalty to ensure a 10-point lead going into the break.
Holwell extended the lead with another penalty shortly into the second half but Glasgow were not finished yet.
Rory Lamont had the home crowd on their feet by breaking three tackles before touching down for a try.
Parks knocked over a difficult conversion to instill hope of a comeback.
The No.10 added a try to his personal tally a couple of minutes later, producing a 50 metre run, and slotting the extras to put his side in controlof the contest.
But Leinster showed their class by dragging themselves back into the game and took the lead once more with a try from winger Kieren Lewis, converted by Holwell.
Another two penalties from Parks added to the tension but the visitors stole the tie with Ben Gissing grabbing his side's third try, which again was converted by Holwell.
|Glasgow Rugby Score Card|
|Leinster Rugby Score Card|