Mark Bennett's 75th-minute try got the Warriors back within two points of the hosts, but Stuart Hogg missed the difficult right-sided conversion as Leinster hung on to reach their fourth successive league final.
The result sets up an all-Irish decider between Leinster and Ulster at the RDS on Saturday, May 25, in what will be Joe Schmidt's final match in charge before he takes on the Ireland head coach role.
Schmidt's charges will not have long to get over their bruising clash with Glasgow, as they take on Stade Francais at the same venue in next Friday's Amlin Challenge Cup final.
The province's enviable record in knockout fare - this was their seventh straight semi-final win in all competitions - helped them overcome a Glasgow side that has been consistently knocking on the door in recent seasons.
Their free-flowing approach under Gregor Townsend was evident in the first half as Niko Matawalu's opportunist try helped them go 10-3 in front.
But a try from man-of-the-match Jamie Heaslip and two Jonathan Sexton penalties, the second after Matawalu was yellow carded, edged Leinster in front for half-time.
Although two more Sexton penalties put Schmidt's side 17-10 ahead, they never looked comfortable and Bennett's score made for a frantic finale which just went Leinster's way.
- Sports & Recreation
- Joe Schmidt