Glasgow edged ahead at half time thanks to a Chris Fusaro try converted by Stuart Hogg in the 39th minute, after Jimmy Goperth's penalty had given Leinster an early lead.
The Warriors then dominated the second half but, due largely to three separate kickers missing six shots at goal, they were dependent on a late try from Jerry Yanuyanutawa to give them the win after Goperth's second penalty cut their lead to just a point.
In perfect conditions both sides looked to be enterprising from the kick-off and it was the visitors who had the better territorial advantage of the first quarter.
However a mixture of tight Glasgow defence and Leinster errors meant the play was rarely within the home 22.
A notable sinner was Gordon D'Arcy who made a dangerous break only to lose the ball when tackled.
There had been little pattern to the scrum penalties and when one came to Leinster after 24 minutes around 35 metres from the Glasgow posts, recent Kiwi recruit Goperth stuck it home.
Just three minutes later, when the Leinster scrum stood up under pressure, Hogg had the chance to bring equality from a wider angle, but he was well off target.
After 32 minutes Hogg missed a penalty from around halfway but the pendulum had swung Glasgow's way with the final five penalties of the half awarded against Leinster.
In the 39th minute minute Glasgow chose to put one to touch and from the line out, fly-half Ruaridh Jackson made a dart to the line where flanker Fusaro found a way over with Hogg converting to let Glasgow go in leading 7-3.
Glasgow dominated possession in the opening minutes of the second half but play was all in their half.
They were awarded a penalty after 52 minutes but Mark Bennett's attempt was short.
Rob Kearney then got first to his side's clearing kick allowing them to get upfield and when Glasgow were caught offside, Goperth's successful kick narrowed the gap to one point.
Glasgow thereafter battered away at a Leinster defence that seemed out on its feet.
Jackson missed a penalty but after 75 minutes replacement prop Yanuyantawa forced his way over. Jackson hit a post with the conversion.