Winger Andrew Conway, flanker Ben Marshall crossed in the first half with the unerring boot of Ian Madigan contributing a further 16 points.
The Dragons, who had an enjoyed a morale-boosting 37-6 victory over Zebre last week, did respond through fly-half Steffan Jones' try and eight points from the boot of Tom Prydie as they trailed 26-13 at the break.
Although tries Shane Jennings and Fionn Carr extended the hosts' advantage after the break, two tries from Prydie and Dan Evans nearly brought the Dragons back into it before centre Brendan Macken's injury-time try brought the breathless action to a close.
Prydie did nudge the visitors in front early on when Leinster were caught going in from the side although Madigan responded soon after and straight from the restart the hosts hit the front.
No-one in the visiting defence could lay a hand on centre Noel Reid before he reached their 22 where the ball was recycled for Conway to wriggle his way over the line. Madigan converted.
However Reid undid much of his good work when his wild pass was picked off by fly-half Jones who had just enough gas to outsprint the fast closing John Cooney.
Prydie added the extras and traded penalties with Maidgan to leave the score at 13-13.
But in the final 13 minutes of the half, Leinster started to pull away as Madigan slotted a 28th minute penalty for an infringement by Ian Nimmo before the home pack marched the ball deep inside the Dragons 22.
With enough defenders sucked in, the ball was spun wide for Marshall to cross unopposed with Madigan converting and the away cause immediately became that bit harder when prop Nathan Williams was binned for infringing at the breakdown.
Some sloppy hands in the home pack prevented Leinster capitalising with a try although Madigan's penalty on the stroke of half-time ensured they had a commanding 13-point lead.
The Leinster forwards redeemed themselves seven minutes after half-time when they formed a maul from a lineout deep inside the Dragons 22 and Jennings had the simple task of touching down.
Madigan converted and three minutes later created the bonus-point score when his brilliant pass released the flying Fionn Carr to dive into the corner to make it 38-13.
The extra point in the bag seemed to inspire complacency in the home ranks as the Dragons hit back with two quickfire tries.
First on 51 minutes, Prydie took an offload before kicking on the afterburners to go over in the corner but the winger was not able to convert his own score.
Nine minutes later, Prydie was again involved as Andy Tuilagi made the initial inroads before the winger set full-back Dan Evans under the posts.
The wind behind their sails, the visitors should have had a bonus point at the very least when Pat Leach failed to gather Toby Faletau's offload with the tryline begging.
But Leinster weathered that storm and re-gathered themselves to have the final word through centre Macken's try in stoppage time that Isa Nacwea converted.
|Leinster Rugby Score Card|
|Newport Gwent Dragons Score Card|