The Dragons hadn’t lost at Rodney Parade so far this season and looked to be on course for victory when they led at half-time.
Lyn Jones’ men had stifled a much-changed Leinster side for almost an hour until Jones was sent to the sin bin on 57 minutes.
With a one-man advantage, Leinster ran in two tries through hooker Aaron Dundon and centre Noel Reid to help seal a win they barely deserved.
After a lacklustre opening 12 minutes, Leinster outside half Jimmy Goppeth kicked the visitors in front with a penalty before Tom Prydie hit straight back for the home side.
Leinster had a try ruled out by the TMO on 27 minutes and it proved to be a big turning point in the first half because four minutes later, the Dragons scored a superb try of their own.
Richie Rees took a quick penalty from five metres inside the Dragons’ own half and great hands from Patrick Leach set Prydie free out wide. He found centre Ross Wardle in support to go over in the corner with Jason Tovey adding the extras.
Gopperth brought Leinster back to within four with a penalty on the stroke of half time.
Tovey kicked an easy penalty within two minutes of the break before Gopperth hit back with three of his own on 50 minutes.
But when the Dragons handed Leinster an advantage when Jones was sent to the sin bin for pulling down Rhys Ruddock, the visitors showed their class.
They choose to kick to the corner rather than take the three points and after several well-worked phases, hooker Dundon powered over to leave Gopperth with an easy conversion.
With their noses back in front, Leinster got a second try four minutes later.
After softening the Dragons’ defence up with some short drives through the forwards, they moved the ball to the blindside where South African Zane Kirchner handed off Prydie and set Reid up for a 30 metre dash under the posts.
But Leinster did not have it all their own way in the final 10 minutes and a yellow card for substitute Jack O’Connell either side of two Tovey penalties set up a grandstand finish.
A late attack left the Dragons five metres from the Leinster line but the visitors held on for the win.
