The tries came early on at Rodney Parade as David Kearney's chip and chase allowed the province to attack early and the Dragons never reacted. Sean Cronin went over in the corner after two minutes and Ian Madigan converted from the touchline.
The Dragons, who edged out Zebre by a single point in the previous round, got a penalty almost immediately from the restart, Steffan Jones showing his accuracy from the kicking tee.
But the floodgates opened in almost the same manner as the first try, Leinster sweeping the ball across from Kearney and quick hands allowed Cronin to touch down for his second try deep in the corner. 12-3 after less than 10 minutes and Leinster looked to be in irresistible form.
The Welsh region were yet to settle with Madigan, Fionn Carr and Kearney's kicking making the home side uncomfortable.
The visitors' pack also weighed in at scrum time, which is where the third try eminated from. Rhys Ruddock retrieved the ball and crashed through the defence, a simple offload later and winger Kearney was over in the corner. Madigan made amends for his earlier miss by confidently slotting home.
The midfield were running the show and not allowing the Dragons any room, although Steffan Jones got his second penalty on the half hour mark.
Jack Dixon was sent to the sin-bin a few moments later which seemed to galvanise the hosts and a Jones drop goal made it 19-9 just before half-time.
On a cool Newport night, the hosts made an instant impression after the restart. The returning Dixon went through the defence and gave the ball to Ieuan Jones to score - the Dragons were suddenly back within five points, and looking confident.
Strong goal-line defence prevented the elusive fourth try for Leinster with the men of Gwent almost breaking free for a try of their own.
Amid a crucial period in the tense Rodney Parade atmosphere as the replacements were wheeled out, Kearney broke the deadlock with his second score.
He doubled his try-scoring tally for the night after good hands in the 22, with Madigan's conversion extending the province's advantage.
It was a lead that the Dragons never challenged, except at the end when Steffan Jones burst through, failing to convert his own try, but it enough to give the Welsh region a deserved point.
Leinster secured the maximum points on St. David's Day in Wales, with aspirations of overhauling the top two, Glasgow Warriors and Ulster, whom they meet next when the league resumes in three weeks' time.