Leinster will play Glasgow in the Pro12 play-offs next Saturday in Dublin after finishing their regular season on top of the table with an 18th league win.
A try from prop Jack McGrath and points from the boot of fly-half Ian Madigan saw Leinster home.
Flanker Dan Lydiate and winger Will Harries scored consolation tries for the Dragons at the end.
The other points came from two Lewis Robling penalties and a conversion by replacement centre Adam Hughes.
However, neither side had a great deal to play for as Leinster had already booked their play-off place as league leaders and the Dragons could only finish mid-table at best.
Leinster's Madigan made the Welsh region pay with three penalties inside the first 17 minutes while Robling countered with two of his own to keep the Dragons within three points of the Irish province.
Dragons skipper Luke Charteris, on his final match before leaving for France, nearly went over for the home side as they stepped up a gear and countered.
But the Wales international second row was held up short, while scrum-half Wayne Evans dashed around the blindside of the resultant ruck only for his try to be ruled out for a Leinster penalty.
It was a lively first half which turned towards Leinster when they grabbed the opening try of the match which was converted.
The pack worked themselves into the home 22 before the ball found centre Noel Reid. His slick hands then gave loosehead prop McGrath his moment of glory as he rumbled over for a try from five metres.
Trailing 16-6, the Dragons had much more of the match at the start of the second half but could not add to their tally.
Instead, Madigan landed his fourth and fifth penalties of the night, while Lydiate scored with four minutes left, with Hughes converting, and Harries came up with the ball under a pile of bodies in the dying seconds.
Newport-Gwent D'gons: Amos, Harries, Hughes, A Smith, Brew, Robling, W. Evans, N Williams, Parry, N Buck, Charteris, Sidoli, Lydiate, L. Evans, Faletau.
Replacements: S. Jones, Price, Way, Groves, Brown, J Evans, Leach, Dixon.
Leinster: D. Kearney, Conway, Macken, Reid, Carr, Madigan, Reddan, J. McGrath, Cronin, White, Toner, Browne, McLaughlin, Ryan, Auva'a.
Replacements: Dundon, van der Merwe, Hagan, Flanagan, Murphy, L McGrath, Nacewa, Hudson.
Referee: Marius Mitrea (Italy)