Edinburgh's Ben Cairns gets to grips with Girvan Dempsey
Leinster's Filipe Contepomi hit the post with a stoppage-time penalty as Edinburgh kept the Magners League title alive for another week.
The visitors, who needed a bonus-point victory to win the league, launched a series of attacks but trailed 10-6 at the interval thanks to a Ross Ford try.
They continued to press but Dave Callam added a breakaway try for the Scots.
Luke Fitzgerald did cross the line for Leinster but Contepomi, who had bagged eight points, failed at the death.
Leinster found it difficult to break down a well-organised Edinburgh side, who were without influential skipper Mike Blair.
The Irish province launched the first telling raid and Contepomi broke the deadlock with a confidently-struck penalty in the seventh minute.
Edinburgh, who had their own title challenge derailed by defeats in Cardiff and Glasgow, replied with a fine touchdown by fit-again hooker Ford.
The Scotland star was in the right place at the right time out wide to finish off a slick handling move.
Phil Godman slotted the tricky conversion, only for Contepomi to double his penalty tally.
However, it took Edinburgh only two minutes to restore their advantage through the boot of Godman.
There was a setback for the Scots in stoppage time, however, as skipper Ally Hogg was yellow-carded for a professional foul.
Leinster stepped up the pace going into the final half hour but Edinburgh scrapped for everything and having soaked up the pressure they delivered a devastating blow when flanker Callam galloped through for the second try.
Fitzgerald revived Leinster's prospects with a pack-drive touchdown, goaled by Contepomi.
Then came high drama deep in stoppage time as Edinburgh were penalised for handling on the deck, leaving Contepomi with a simple strike which crashed back off the woodwork.
Edinburgh: Southwell, Webster, Cairns, De Luca, Houston, Godman, Laidlaw, Allori, Ford, Kerr, Mustchin, Gissing, Hogg, MacDonald, Callam. Replacements: MacRae for Houston (29), Meyer for Laidlaw (71), G. Cross for Kerr (41). Not Used: Kelly, Hamilton, Newlands, D. Blair.
Leinster: Dempsey, Fitzgerald, O'Driscoll, Contepomi, Kearney, Sexton, Whitaker, le Roux, Jackman, Wright, Cullen, Hogan, Keogh, Jennings, Heaslip. Replacements: Warner for O'Driscoll (67), O'Kelly for Cullen (36), Gleeson for Jennings (55). Not Used: Blaney, Knoop, Keane, Brown.
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