The defeat heaped further woe on lowly Warriors who have now lost six games in succession since their 26-20 win over Munster at Thomond Park on 18 February.
The hosts, who produced one of their better displays of the season, took the early lead with a long-range penalty from Dan Parks but Leinster' Felipe Contepomi cancelled that out with one of his own soon after, having missed his first opportunity to do so.
Glasgow's hopes took a blow when Andy Henderson was sin-binned for pulling back Guy Easterby off the ball and moments later Keith Gleeson powered through for the Irish side's first try after Glasgow prop Kevin Tkachuk had spilled the ball on his 22.
Contepomi converted to cement his side's lead.
Leinster took a stronger hold on the game on thehalf hour mark when Brian O'Driscoll provided a well-timed pass for wing Robert Kearney to go over in the left-hand corner for the second try of the game.
Glasgow raised their game after the re-start and were rewarded when Henderson snaffled an interception before racing away for the touchdown.
Parks converted to close the deficit to five points but another penalty from Contepomi kept Leinster, who by now had lost O'Driscoll to injury, ahead in the contest with a simple penalty.
Parks misse dhis next shot at goal and there was another blow for Glasgow on the hour mark when skipper Donnie Macfadyen was yellow carded for illegally handling on the ground at a ruck.
As the game entered the final quarter Parks and Contepomi exchanged further penalties before replacement prop Stuart Corsar raised hopes of a late comeback with his side's second try.
However Leinster held on for the win and in doing so kept their push for Celtic League glory on course.
|Glasgow Warriors Score Card|
|Leinster Rugby Score Card|