David Kearney dived over for the province's only try after 66 minutes to tee up a title decider against the Ospreys at the RDS on Sunday, May 27, but not before the Scotsmen had touched down twice in the last three minutes through Dougie Hall and Stuart Hogg.
Jonathan Sexton struck four penalties and a conversion to bolster Leinster's tally, with Glasgow with one penalty and a conversion.
The final is now due to be a repeat of the 2010 edition, which the Ospreys won, and for Leinster it could be the second piece of a league and European double.
With next week's Heineken Cup final in mind, Leinster again rejigged their starting line-up with Brian O'Driscoll (knee) and Rob Kearney (back) both resting injuries.
But the hosts exerted the early control, retaining possession through a series of rucks before Sexton slotted a seventh-minute penalty.
Glasgow were level only two minutes later thanks to a pinpoint kick from Duncan Weir and the visitors drew encouragement from their rugged defence with their scrum also shaping up well.
Sean Lineen's men made Leinster work hard for any territory and Richardt Strauss was penalised for not releasing close to the Glasgow try-line.
Shortly afterwards Sexton restored the hosts' lead with a penalty from just outside the 22, and a clever kick-chase from Isa Nacewa drew the crowd's approval.
After an earlier warning, Alex Dunbar was dispatched to the sin-bin in the 28th minute for blatantly coming in at the side of a ruck five metres from his line.
Federico Aramburu knocked on when trying to intercept a Leinster pass as Glasgow dug their heels in, successfully surviving some hairy moments in their 22.
Fergus McFadden and Cian Healy both went close to breaking the try deadlock before Weir lifted the siege by launching turnover ball downfield.
Back came Leinster though and they gleaned three points from their numerical advantage courtesy of a 38th minute strike from Sexton, going in 9-3 at the break.
Weir missed a chance to open the scoring in the second half, pushing a difficult penalty from distance to the right of the posts.
The game remained tight and tense and Leinster were forced into their second change at centre with the limping Eoin O'Malley joining a bloodied Gordon D'Arcy on the sidelines.
Weir was off target with a penalty attempt from the left and a powerful Leinster scrum robbed Glasgow of possession and a promising position in the home 22.
Despite a penalty miss from Sexton, Leinster turned the screw around the hour mark with Brad Thorn and Shane Jennings ensuring quick ruck ball for them.
Normal service was resumed when Sexton popped over his fourth penalty for a 12-3 scoreline and a rare chink in the Glasgow rearguard allowed Kearney to grab that all-important try.
Jennings put Nacewa through on a diagonal run and he passed for Kearney to make it over in the left corner. Sexton added the extras from the touchline to heighten the blow for Glasgow.
The Warriors showed no signs of wilting though and after Andrew Conway had a try ruled out for an earlier infringement, the Scots closed the gap.
The ball was spun wide for replacement hooker Hall to crash over in the right corner and in injury-time, Ruaridh Jackson's neat grubber kick was gobbled up by Hogg for a second consolation score.