Glasgow were rocking now and Leinster were soon across the line again as prop Clarkson powered over despite the attentions of a clutch of Warriors off the back of a scrum.
Sheehan added another try on the stroke of half-time, powering over behind a ruck following a line-out close to the line, to give the Irish side a 20-point advantage at the break.
The gap would have been wider had Prendergast been more accurate with the boot, but with Glasgow pinned back in their own half after the restart, the fly-half made no mistake after Osborne dived over for his second try near the posts.
Replacement fly-half Frawley added another try in the corner before weaving finishes from Dobie and Tuipulotu made the scoreline more respectable for the visitors.
Leinster head coach Leo Cullen: “We reacted really well to Glasgow’s early score. I think we managed the game and the conditions very well.
“It’s all about getting through no matter how we did it, but we made the right decisions throughout the game. We prepared well this week and we will do so again this week.
“We will try to make the travel that one of the South African teams will have to make next week count.”
Glasgow Warriors head coach Franco Smith: “They are a quality outfit and brought a huge performance. I felt their urgency and they put pressure on us.
“On the day, we were not good enough, especially in the first half. We should have started better and then we wouldn’t have had to chase the game. I don’t think the strategy was wrong, they are good at the aerial battle, but I am really proud.
“We had a lot of boys unavailable this week. I must look at the big picture. It was a tough task to come here and win.”
Leinster: Jimmy O’Brien; Tommy O’Brien, Jamie Osborne, Jordie Barrett, James Lowe; Sam Prendergast, Jamison Gibson-Park; Andrew Porter, Dan Sheehan, Thomas Clarkson, Joe McCarthy, James Ryan, Ryan Baird, Scott Penny, Jack Conan (capt).
Replacements: Ronan Kelleher, Jack Boyle, Rabah Slimani, RG Snyman, Max Deegan, Luke McGrath, Ross Byrne, Ciaran Frawley.
Glasgow Warriors: Josh McKay; Kyle Steyn (capt), Sione Tuipulotu, Tom Jordan, Kyle Rowe; Adam Hastings, George Horne; Jamie Bhatti, George Hiddleston, Fin Richardson, Alex Samuel, Scott Cummings, Euan Ferrie, Rory Darge, Henco Venter.
Replacements: Johnny Matthews, Rory Sutherland, Sam Talakai, Max Williamson, Jack Mann, Macenzzie Duncan, Stafford McDowall, Jamie Dobie.
Referee: Andrea Piardi (Italy).
