The five-try fillip leaves the reigning champions in contention to reach the quarter-finals, ahead of the second part of their rescue mission away to Exeter next Saturday.
The home side had only scored three tries in their four previous pool games, and the loss of Sean O'Brien to illness before kick-off did not bode well.
But first-half touchdowns from Cian Healy, Shane Jennings and man-of-the-match Luke Fitzgerald saw Leinster sweep into a 19-6 lead at the RDS.
Scarlets full-back Liam Williams snuck in for a five-pointer before the break, only for his opposite number Rob Kearney to reply with the all-important bonus-point score.
Ian Madigan came off the bench to grab a fifth try in the final minute, as the Irish province's focus switches to that decisive clash in Devon where they need another bonus point victory.
Winless in Europe this season, the Scarlets came to Dublin looking for a January jolt amid a spate of injuries and deflating defeats - 30 points separated them from Ulster last weekend.
Aled Thomas landed the opening points from a third minute penalty set up by a powerful carry from Sione Timani, a late addition to the visitors' back row for an ill Kieran Murphy.
Leinster turned down a shot at the posts as they hunted for the game's first try, using the lineout to launch a series of close-in carries.
The move paid dividends as prop Healy used ruck ball to worm his way over from a metre out. Sexton quickly converted as Leinster gained early momentum.
A snap drop-goal from Liam Williams, who was fed on the right outside the 22, made it a one-point game. Leinster's response was a familiar one, shunning a penalty kick in favour of going up the line.
Leinster's usually reliable lineout let them down initially, but a neat midfield link-up between Sean Cronin and Sexton got the hosts back firing.
Then set piece ball closer in provided the platform for the maul from which flanker Jennings muscled over for try number two - Sexton missed the right-sided conversion.
Leinster's improved accuracy with ball in hand saw them swiftly create another try-scoring chance, their pack giving the Scarlets little time to breathe as Healy and Fergus McFadden both went close.
The Llanelli men leaked a scrum penalty and the tackle count took its toll as Healy offloaded for Devin Toner to send the fit-again Fitzgerald in at the left corner.
It was the winger's first European outing since last season's semi-final - he had to undergo neck surgery in May - and Sexton followed up with the conversion.
But the Scarlets got an ideal response in first half injury-time with Liam Williams going over unchallenged after Scott Williams delivered a crafty cross-field kick.
Leinster though were ominously direct right from the off in the second period and made certain of the bonus point just four minutes in.
Heaslip peeled away to the right from a scrum in the 22, with Kearney collecting his offload, breaking the first tackle and spinning out of a second to score under the posts.
Sexton converted for a 26-11 scoreline and with Gordon D'Arcy and Fitzgerald growing in influence, the province continued to plug away for scores.
The tough tackling Timani led the Scarlets' resistance and Thomas kicked a 57th minute penalty as the visitors remained competitive.
In a ragged final quarter, Leinster slackened in terms of creating scoring chances as the Welshmen defended doggedly to the last.
But the home side's patience paid off late on as Madigan's quick feet took him past two defenders, and he drop-kicked the conversion to complete the scoring.