A try from prop Jack McGrath and points from the boot of fly-half Ian Madigan saw Leinster home.
Flanker Dan Lydiate and Will Harries scored consolation tries for the Dragons at the end, adding to a couple of Lewis Robling penalties and a conversion from 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.
In what was their last ever game at Firhill Stadium before their summer move to the recently redeveloped Scotstoun Stadium, the Warriors impressed throughout and grabbed the spoils thanks to tries from DTH van der Merwe and John Barclay.
The rest of the points came from the boot of Duncan Weir, while replacement Ruaridh Jackson also chipped in with a late conversion. Despite their loss, Connacht remained in eighth place in the table, their highest league finish ever.
Knowing their season had already been extended, there was little pressure on the Welsh side, but they still rose to the occasion and produced a win that gives them some momentum heading into the play-offs.
On a tight night, the boot of Dan Biggar proved decisive, as he kicked two penalties and also converted one of his side's two tries which were scored by Richard Fussell and Hanno Dirksen.
Simon Zebo and forwards Peter O'Mahony, Mike Sherry, Stephen Archer and Tommy O'Donnell all crossed for the defending league champions, who will travel to the Ospreys in next weekend's semi-finals.
However, the Thomond Park success was marred by a suspected knee injury which forced Paul O'Connell off midway through the second half.
In the day's other games