Wing Isa Nacewa scored the visitors' only try with Jonathan Sexton supplying 15 points with the boot.
The Scarlets, who came away with a losing bonus point, scored a try through Gareth Maule and two penalties and a conversion from Rhys Priestland.
It gave Leinster their 17th successive victory in the tournament and coach Joe Schmidt's 17th win from 19 Heineken Cup games.
Irish centre Gordon D'Arcy returned as Leinster made two changes from the Exeter clash - Sean Cronin coming in at hooker and Fergus McFadden switching from centre to wing.
The Scarlets were without Wales centre Jonathan Davies (groin) but in-form wing Andy Fenby returned.
Maule replaced Davies to partner Scott Williams in midfield while full-back Liam Williams and flanker Aaron Shingler recovered from knocks sustained in the 49-16 defeat in Clermont Auvergne.
Leinster were given a great start with Sexton kicking a second minute penalty goal from the 22 after the Scarlets killed the ball at a ruck. It took Sexton past 350 points for the tournament.
The Welsh side were moving the ball intelligently behind but gave away another penalty as Sexton was able to kick the Irish side into the home 22.
Leinster gained another penalty from which they built territory and scored a try through Nacewa, who collected a cross-kick from Sexton despite having George North bearing down on him.
Sexton missed with the conversion but then his opposite number Priestland was wayward with a long-range penalty on 16 minutes.
And Sexton was back on target as the defending champions took an 11-0 lead with a penalty after the Scarlets had been caught offside.
The Scarlets just could not cope with Leinster's streetwise approach at the breakdown and despite a later surge towards the line the Welsh side ended the first half pointless.
The Scarlets had the chance to get on the scoreboard within two minutes of the restart but Priestland was well short with a 45-metre effort.
But it proved to be a double whammy for the home side as a minute later Sexton put over an opportunist drop-goal from fully 40 metres.
The Scarlets finally broke their duck in the 48th minute with Priestland kicking a penalty after Leinster had failed to release in the tackle.
To the delight of the home fans, Sexton was off target with his own penalty three minutes later.
But the home fans were sent into raptures when a long pass from Priestland sent Gareth Maule on an outside break to the right corner leaving O'Driscoll in his wake. Priestland converted the try as Scarlets closed the gap to four points.
That was extended to seven as Sexton eased some of the second half pressure that had been building on Leinster with his third penalty.
Sexton and Priestland swapped penalties as Leinster led 20-13 with five minutes remaining, but they ended the match down to 14 men with full-back Ian Madigan in the sin-bin for taking opposite number Williams in the air.
Sexton was wayward with a 79th-minute penalty, which would have brought the Scarlets a bonus point had the kick gone over the posts.
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