A tight opening half saw both sides fight doggedly for territory but it was Leinster who broke the deadlock after 10 minutes with a try from Isa Nacewa.
© PA Photos
Scarlets produced a stirring second-half fightback and scored a fine solo try through centre Gareth Maule.
Jonathan Sexton maintained Leinster's advantage with a series of efficient penalties and a drop goal from range.
Scarlets were given a lifeline with ten minutes to go when Ian Maddigan was sinbinned for a high tackle, but could not make the most of their man advantage.
A composed performance from Leinster made it two wins from two for the defending champions and puts them top of Pool 5 with eight points.