Adam Hughes used turnover ball to manufacture a superb 19th minute try and eight points from the boot of Jason Tovey had the Dragons 13-10 ahead at the break.
Tovey kept the Welsh region in the hunt at 16-16, but Leinster produced the stronger finish to move into second place ahead of next weekend's trip to leaders Munster.
Speaking after the game, the province's head coach Joe Schmidt said: "We're satisfied to get the win because we knew how tough the Dragons would be.
"I thought that they were unlucky against Ulster last week and they were a bit unfortunate at times today as well. We didn't look too cohesive today, which was frustrating.
"It was great to get the win after the Ospreys slipped up in Edinburgh, while Munster did us a favour against Cardiff on Friday night. So we have a bit of a buffer (at the top) but we have to go to Munster next week and that will be another massive challenge."
Sexton got Leinster up and running with a third minute penalty after man-of-the-match Isa Nacewa and a hungry Luke Fitzgerald had featured in attack. Their lead remained intact when Tovey missed a kickable effort, three minutes later.
Sexton was narrowly off target with a long range penalty, before a well-timed tackle by Aled Brew on the advancing Fitzgerald saw the ball run loose in the Dragons' half.
Joe Bearman scooped it up for Hughes, on the left wing, to set off on an opportunist 60-metre run which saw him outpace Shane Horgan for the try-line. Tovey added the conversion but the Dragons' lead was quickly cancelled out.
Sexton knocked a close range penalty to touch, the forwards drove up to the line and Stan Wright went close before scrum half Isaac Boss lunged over from metres out. Television match official Tony Rowlands confirmed the grounding.
Sexton converted to draw Leinster level at 10-10, only for Tovey to reply two minutes later and give Darren Edwards' side a slender advantage for the second half.
Hughes almost snuck over for his second try on the resumption. The early pressure came from the Dragons, although Tovey's third penalty success was their only reward.
Two quick-fire penalties from Sexton had given the visitors a brief lead. His second strike came after Dragons winger Brew was sin-binned for a trip on Nacewa.
Horgan looked to have stretched over for a try in the right corner as Leinster began to string some phases together, but Rowlands ruled it out and the Dragons survived.
That was until Leinster's power up front, aided by replacements Nathan Hines and Jamie Heaslip, forced the hosts backwards in a couple of scrums.
After a wheeled effort to the right of the posts, referee Andy Macpherson awarded a penalty try which Sexton comfortably converted.
Sexton extended Leinster's lead to 10 points with a 45-metre penalty on the hour mark, but the Dragons continued to threaten from deep and around the fringes.
Their top try scorer this season, Brew, was a nuisance for the Leinster rearguard to keep tabs on. He supported Toby Faletau on a promising breakaway, but the young number 8's pass was too strong and was fumbled.
Despite losing flanker Dominic Ryan to the sin-bin, Leinster finished in good stead. A midfield surge from Kevin McLaughlin got them firing again, and replacement David Kearney was bundled into touch by Hughes as he hunted for a late try.
|Newport Gwent Dragons Score Card|
|Leinster Rugby Score Card|