Dougie Fife claimed the only try of the match as Edinburgh ran out 11-6 winners , with Greig Laidlaw's two penalties also proving crucial.
Leinster, whose points arrived via two penalties from the boot of Ian Madigan, applied pressure late on but they could not avert another setback following their home defeat by Northampton in the Heineken Cup, and they slip to third in the standings.
Edinburgh kicked off amid a downpour that had been falling for several hours and in a powerful wind that swirled and changed direction, ensuring neither team was favoured.
Jack Cuthbert and Tonks kicked effectively early on as Edinburgh applied the pressure, but poor handling and mistakes at the line-up stifled their attacks.
Despite their bright start the hosts conceded a penalty in the 21st minute for failing to release and Madigan was able to kick the visitors in front.
The lead did not last long, however, with Laidlaw soon scurrying 30 metres after being released by Ollie Atkins. Although he was brought down on the visitors' 22, Edinburgh kept possession and a smart pass from Tonks sent wing Fife sliding in on the left.
Edinburgh remained on the attack through to half-time and went in 8-3 ahead thanks to a straightforward penalty from Laidlaw, who had failed with his earlier conversion attempt.
Soon after the restart Madigan kicked his second penalty to reduce the arrears to two points, but within three minutes the lead was back out to five as Laidlaw successfully sent over another three-pointer .
The game descended into a scrappy kicking contest as the miserable conditions took their toll, although on 73 minutes Edinburgh's forwards drove forward a mini maul to win a penalty but Laidlaw could not take the opportunity to kick his third of the contest.
Leinster were never out of the contest by any stretch and found belief in the final minutes, so nearly breaking through for a match-winning try.
Following a line-out they took up camp on the home line for a series of thrusts as the clock ticked down but they were all repelled by the formidable Edinburgh defence, with Leinster's hopes finally dashed by a late and costly knock-on.
- Sports & Recreation
- Greig Laidlaw
- Ian Madigan