Michael Allen touched down early for the hosts but Dan Leavy, Garry Ringrose, Isa Nacewa and Jamison Gibson-Park responded emphatically.
Edinburgh improved greatly after the interval, with Hamish Watson ploughing over from close range.
Magnus Bradbury's late try brought Edinburgh back to 26-20, only for Leavy to add a second score.
Nacewa's last-minute conversion, his fourth of the match, took him to 500 points for last year's beaten Pro12 finalists.
Much of the debate in the aftermath of Edinburgh's stodgy victory over Scarlets last week centred on the supposed lack of attacking ambition from Alan Solomons' men, but they needed just 90 seconds to tear through the Leinster defence.
Salsa Tofilau slashed through the midfield defence and when the ball was recycled and shipped wide to Phil Burleigh, he weighted a delightful grubber kick for Allen to touch down.
Duncan Weir added a penalty to stretch the lead to 8-0, but Edinburgh's defence, so impressive in shutting down Scarlets' last week, left a gaping hole for Leinster open-side Leavy to gallop through and coast under the posts to make it a one-point game.
The variety of the Leinster attack was now proving problematic for the home side to contain, and when Ringrose and Nacewa crossed in quick succession, the momentum had shifted completely.
The threats to the Edinburgh line were coming from all over the pitch and, with the home side pinned on their own line, scrum-half Gibson-Park darted over to make it 8-26 at the break.
Watson's bulldozing run at the start of the second period suggested Edinburgh were not prepared to go down without a fight, and the open-side was rewarded for his belligerent display when he ploughed over for a try that was converted to offer hope.
The home side enjoyed an extraordinary 81% of possession and 82% territory in the second-half, but despite promising breaks from the likes of substitutes Blair Kinghorn and Solomoni Rasolea, when cool heads and crisp passing was required to finish off the chances, both were absent.
That was until five minutes from time when good work from another substitute, Fraser McKenzie, allowed Bradbury to burst through to bring his side back within six points.
The losing bonus point would have been just reward for Edinburgh's display after the break, but Leavy powered over at the death to deny them even that consolation.
Edinburgh coach Alan Solomons: "Our defence wasn't good and I think that ultimately cost us the game. Defensively, I thought we were really poor.
"At 26-8, we were in real trouble. We regrouped well, we had a lot of the ball and a lot of territory in that second-half.
"Little errors cost us and we didn't translate that pressure into points.
"At 26-20 and with every prospect of pulling off a win, poor defence on our line saw that bonus point slip away. That was really upsetting.
"I thought the regrouping after half-time was very good because we were under quite a lot of pressure, but defensively we were poor and the error rate was too high."
Edinburgh: Glenn Bryce, Michael Allen, Sasa Tofilau, Phil Burleigh, Tom Brown, Duncan Weir, Nathan Fowles, Rory Sutherland, Ross Ford, WP Nel, Ben Toolis, Grant Gilchrist (capt), Magnus Bradbury, Hamish Watson, Viliami Fihaki.
Replacements: Stuart McInally, Allan Dell, Kevin Bryce, Fraser McKenzie, Cornell Du Preez, Sam Hidalgo-Clyne, Solomoni Rasolea, Blair Kinghorn.
Leinster:  Zane Kirchner, Isa Nacewa (capt), Garry Ringrose, Noel Reid, Dave Kearney, Joey Carbery, Jamison Gibson-Park, Jack McGrath, Sean Cronin, Mike Ross, Devin Toner, Mike McCarthy, Jordi Murphy, Dan Leavy, Jamie Heaslip.
Replacements: Bryan Byrne, Cian Healy, Tadhg Furlong, Ross Molony, Josh van der Flier, Luke McGrath, Cathal Marsh, Rory O'Loughlin/
Leinster earned a bonus point win at Murrayfield, exploiting Edinburgh's shaky defence in the first half.