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The visitors won with an injury-time penalty from out-half Brendan Laney following the dismissal of centre David Quinlan for foul play.
Gary Ella’s side were 14 points in front in the final quarter, before a Laney-inspired Edinburgh scored two late tries to level by the 84th minute.
Ella’s charges got off on the right foot with former England Under-21 fly-half Matthew Leek flashing over a seventh minute penalty, 15 metres in from the touchline.
Missing the services of Australian centre Christian Warner, out with a neck muscle strain for 14 weeks, the home side took a foothold in Gunners’ territory and two minutes later saw scrum-half Brian O’Riordan scamper over for the game’s first try, from the base of a scrum inside Edinburgh’s 22.
Leek converted before Scottish international Laney landed a 10th minute penalty in reply.
An alert Gordon D’Arcy side-stepped past Derrick Lee for his first try of the season before the midpoint of the half. But out-half Laney, funnelling behind a slow-starting pack, ghosted over with three minutes remaining to close the interval gap to a single score at 17-10. Then a fourth successful kick from Leek gave Ella’s men a 20-10 by the second half’s seventh minute.
D’Arcy’s second of the night from a flat pass from wing John McWeeney put Leinster 14 points in front by the hour mark.
Laney’s 51st minute drop goal was the sole reply for Frank Hadden’s side, before full-back Lee and centre Tom Philip crashed over to reduce the deficit.
D’Arcy’s sin-binning for a deliberate ruck offence saw Leinster endure a cagey period. Memories of Edinburgh’s 1999/2000 European Cup comeback from 17 points down against the Irish province for a draw looked to be banished with O’Meara’s 81st minute penalty, but cruelly for Leinster, Craig Joiner took an inside pass to skip over in injury time.
Laney converted, and with Quinlan red-carded for an off-the-ball offence by Welsh referee Hugh Watkins, the Scottish international kicked Edinburgh to top spot in the Celtic League table, and consign Leinster to rock bottom.