Rhys Thomas touched down for the Scarlets, who had the final say with a late consolation try from replacement Nic Reynolds.
The game pitted the league's third against fourth-placed sides against each other and, as such, it was no surprise that the first half was a cagey affair.
With a neck injury ruling Iestyn Thomas out, Gareth Maule captained the Scarlets and he knocked on as the visitors looked to create an early opening in the Leinster 22.
Rhys Priestland dragged a drop goal wide after a turnover from Josh Turnbull, but it was Leinster who managed to open the scoring through winger Morris.
He was invited forward on the right wing and fended off a poor tackle from Priestland to gallop over in the corner. Sexton converted for good measure.
The Scarlets were eager to put the ball through the hands and teenager George North saw a good deal of possession, even if Leinster were quick to close him down.
Priestland had two penalty misses either side of a successful 20th minute shot from Sexton, before he redeemed himself near the half hour mark.
It closed the gap for the Scarlets to 10-3 at half-time, but Leinster looked solid with Sexton and Isa Nacewa putting in some clever kicks.
Leinster grabbed the lion's share of territory and possession in the early stages of the second half and the Scarlets let their discipline slip, with Sexton clipping over successive penalties in and around the 22 and a third from the 35-metre range.
Replacement scrum half Gareth Davies kept the Scarlets pack plugging away and their endeavour was rewarded with a close range score from prop Thomas, which television match official Kevin Beggs confirmed.
Number 8 Ben Morgan, a strong carrier throughout for the Scarlets, did well to break towards the line and although he was halted, Thomas had plenty of support to burrow over.
But when Leinster barged back into the Scarlets' 22, successive penalties led to Maule being sin-binned for offside.
The hosts wasted little time in putting the game beyond the visitors' reach as Boss shrugged off Davies' attempted tackle to score to the right of the posts, and Sexton's conversion sealed it.
The winning margin was cut to 11 points when Reynolds nipped over in the left corner for his first try for the region, following some precise passing from Priestland and the impressive Regan King.
Fergus McFadden and young back rowers Dominic Ryan and Rhys Ruddock put in strong showings for Leinster, who brought on former Connacht and Munster back Daniel Riordan late on for his first cap.
Speaking afterwards, McFadden said: "The game was quite patchy at times and in the first half especially we coughed up a fair amount of ball. The Scarlets throw the ball around well and they got two tries which we wouldn't be happy with.
"We knew about their strengths coming into the game and they're a very dangerous side, so we knew that we had to take whatever chances came our way.
"At the end of the day though, we won and we denied them a bonus point - four points is four points."
|Leinster Rugby Score Card|
|Scarlets Score Card|
|Rhys M. Thomas||1||5|