Tuipulotu was replaced at half-time after an impressive cameo as he works his way back to full fitness.
It was a scrappy affair in the second half and Cullen's side were sloppy after the restart, with a number of accuracy errors, although did edge further in front through a Prendergast penalty.
Glasgow hit back a minute later for their first points of the evening, as replacement Horne grounded the ball after Sebastian Cancelliere had poked it through to him, but Hastings missed the conversion.
Leinster replacement Frawley made sure of the win by kicking a late penalty after Glasgow were penalised for an offside.
Leinster Rugby: Keenan; T O'Brien, Henshaw, Barrett, J O'Brien; Prendergast, Gunne; Porter, Sheehan, Clarkson, Snyman, Ryan, Baird, Penny, Conan (capt).
Replacements: Kelleher, Boyle, Furlong, Mangan, Deegan, McGrath, Frawley, Osborne.
Glasgow Warriors: Rowe, Cancelliere, Jones, Tuipulotu, Steyn, Hastings, Afshar; Bhatti, Stephen, Richardson, Oguntibeju, Samuel, Duncan, Darge, Ferrie.
Replacements: Hiddleston, McBeth, Walker, Du Preez, Cummings, McDowall, Horne, Jordan.
Referee: Ben Whitehouse (WAL)
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