Last year's champions led 17-10 at half-time with Noel Reid and Jack Conan scoring either side of Hamish Watson's impressive try for Edinburgh.
Zane Kirchner extended Leinster's lead before Tim Visser's score and Sam Hidalgo-Clyne kicked the hosts ahead.
But tries from Jordi Murphy and Luke McGrath, allied with solid Ian Madigan kicking, meant the Irishmen prevailed.
Leinster end their campaign fifth, seven points shy of the semi-final places.
Edinburgh finish eighth for the second consecutive season under head coach Alan Solomons, but with a total of 48 points this campaign compared to 38 last year.
Add to that a European Challenge Cup final - albeit a defeat to Gloucester - and it seems the South African is making gradual progress at Murrayfield, where he recently signed a new 12-month contract.
Hidalgo-Clyne's early penalty set Edinburgh off on the right foot but the visitors soon established a lead they would hold onto for the most of the match.
A nice line-out collection by Leinster flanker Murphy presented Madigan with a drop-goal opportunity that the fly-half duly took.
Madigan then combined well with Sean O'Brien to release Reid through the middle; the 24-year-old centre outpacing Edinburgh full-back Jack Cuthbert to cross between the posts.
The conversion was added by the boot of Madigan but Edinburgh ensured the Irishman did not go out of sight.
After Dougie Fife's claim for a try in the corner was ruled out, Watson found himself in possession from the line-out and gambled with a darting run to the line instead of spreading the play left were the hosts had numbers waiting.
Watson, who won his first Scotland cap in this year's Six Nations, managed to get behind the posts and Hidalgo-Clyne converted.
However, Leinster were quick to put on another display of quick hands and fast offloads in the middle to double their try count. This time Dave Kearney and Madigan gave substitute Conan the opportunity to split Edinburgh's defence with a piercing run and try, with Madigan again adding another two points.
Hidalgo-Clyne heartened the hosts at the start of the second half with another penalty before Springbok winger Kirchner crossed on the right-hand side for Leinster, although there was no conversion on this occasion to compliment it.
The Murrayfield men's resolve still existed and a fitting try for the Harlequins-bound Visser, who wrestled free to touch down, gave Edinburgh's record try scorer a final moment to remember in red and black.
Hidalgo-Clyne's conversion and subsequent penalty put Edinburgh ahead for a few dreamy moments as thoughts of seventh place and a European play-off became a reality once more, but it was too much to ask of an injury-depleted side.
Murphy and McGrath both ended a period of stout home defending to cross over late on and Madigan added the extras each time.
Edinburgh: Cuthbert, Fife, Beard, Strauss, T. Visser, Heathcote, Hidalgo-Clyne, Dickinson, Ford, Andress, Bresler, Toolis, McKenzie, Watson, Du Preez.
Replacements: Brown for Beard (59), Fowles for Hidalgo-Clyne (78), Sutherland for Dickinson (76), Cochrane for Ford (63), Berghan for Andress (38), Atkins for Bresler (57), Hilterbrand for McKenzie (73). Not Used: Te Rure.
Leinster: R. Kearney, Kirchner, O'Shea, Reid, D. Kearney, Madigan, Boss, J. McGrath, Strauss, Ross, Toner, M. McCarthy, J. Murphy, O'Brien, J. Heaslip, Denton.
Replacements: Kelleher for Kirchner (80), Gopperth for Reid (74), L. McGrath for Boss (65), Bent for J. McGrath (61), Dundon for Strauss (78), Denton for M. McCarthy (3), Conan for Denton (8). Not Used: Furlong.
Ref: Nigel Owens (Wales)
Att: 3,476 Ref: Nigel Owens (Wales)
Edinburgh's flickering hopes of staying in European Champions Cup contention were extinguished by Leinster.