Johnny Sexton contributed 23 points as Leinster emerged victorious from a hard-fought encounter in the Scottish capital.
The Ireland fly-half kicked six penalties, a conversion and a drop-goal to add to tries from Isaac Boss and Leo Auva'a.
The home side edged the try count 3-2, with Tim Visser notching a double and Matt Scott also touching down, and it was well into the final quarter before Leinster finally shook off a determined home side with Fergus McFadden's last-minute penalty easing any remaining nerves.
The home side set the scoreboard ticking after four minutes. Rob Kearney fumbled a kick ahead by Edinburgh's teenage stand-off Harry Leonard and, when the visitors offended at the ensuing scrum, home skipper Greig Laidlaw slotted the kick to edge his side into the lead.
Kearney soon atoned for that error by bursting through the home defence. He was thwarted by a magnificent last-ditch tackle from Visser just short of the line but an offence by Allan Jacobsen at the ruck allowed Sexton to square matters.
And the Leinster stand-off was handed the simplest of tasks on 15 minutes when Edinburgh offended in front of their posts and he stroked over the kick.
Edinburgh struck back instantly when Laidlaw picked up at a ruck on the edge of the Leinster 22.
He fed Visser, who broke the first tackle and evaded the final defender then plunged over between the sticks to leave Laidlaw a simple conversion.
Sexton trimmed the deficit to a single point when he booted his third penalty, then added the conversion after Boss had dummied his way over from a close-range ruck. A further penalty apiece for Laidlaw and Sexton left the visitors 19-13 ahead at the interval.
Edinburgh were in front again within three minutes of the restart, with Visser again the man who did the damage. He took a pass from Chris Paterson just inside the Leinster half and rounded Dave Kearney before hammering through attempted tackles by Rob Kearney and Sexton to claim his second try of the night. Laidlaw's conversion took the home side a point clear.
Sexton restored Leinster's lead with another penalty, but Laidlaw once again responded in kind to give Edinburgh a 23-22 advantage that lasted less than three minutes.
Replacement prop Richardt Strauss thundered into contact and the recycled ball was worked wide to Rob Kearney who released the ball as he fell, finding Auva'a who dotted down in the corner.
Sexton failed with the conversion attempt and also missed a penalty, but normal service was resumed in the 66th minute when he stroked over his sixth penalty of the night and added a drop-goal two minutes later.
Edinburgh were not yet finished and Matt Scott darted onto a clever chip ahead by Leonard to score. However, Laidlaw pushed the conversion wide of the target and McFadden's injury-time penalty robbed the hosts of a losing bonus point.