Fergus McFadden's last-minute penalty denied Edinburgh a deserved bonus point after Tim Visser had scored two tries, and Matt Scott another for the hosts.
Edinburgh took an early lead through scrum-half Greig Laidlaw after Leinster were penalised for collapsing a scrum.
But the reigning European champions were back level after seven minutes as Rob Kearney was hauled down just short of the line by Visser after some good interplay, only for Allan Jacobsen to come in from the wrong side, with Sexton, on his return from the World Cup, adding the three points.
Luke Fitzgerald then made a fantastic half break to put scrum-half Boss away, but the New Zealand-born Irish international didn't quite have the pace to finish off the move and was stopped short of the line before Edinburgh again infringed, leaving Sexton a simple shot at goal from in front of the posts.
Edinburgh's reaction was immediate as Laidlaw laid a beautiful pop pass to Visser who shrugged off Sean Cronin's weak tackle before rounding Rob Kearney to go over, Laidlaw adding the extras to make it 10-6 after 17 minutes.
Virtually from the restart Edinburgh were again penalised and Sexton knocked over his third penalty of the evening.
Leinster reclaimed the lead on 24 minutes as Boss darted off the back of a ruck and slipped past Harry Leonard's tackle to cross, Sexton's conversion making it 16-10.
Laidlaw brought his side back to within three points as he knocked over another penalty after Leo Cullen had been penalised for offside at a ruck but Sexton was just as accurate a minute later for Leinster.
Three minutes into the second half Visser grabbed his second of the night as he went round David Kearney's poor attempt at a tackle before stepping his brother Rob, Laidlaw's conversion putting the home side back in front 20-19.
Sexton put Leinster back in front from wide out on the right but Laidlaw matched his kicking feats with a beautiful kick after Cullen had again infringed.
Leinster surged back into the lead as Auva'a finished off a fantastic move in the corner after replacement hooker Richardt Strauss had made a rampaging break for the visitors. Sexton finally missed a kick from on the right touchline but the visitors led 27-23 on the hour mark.
Leinster should have scored their third try when Rob Kearney broke through the Edinburgh defensive line before finding Sexton whose poor pass was knocked on by Fitzgerald with the line begging.
However Sexton made amends shortly after as he stroked over another penalty and then knocked over a simple drop goal from straight in front to take his side to a 33-23 lead.
With eight minutes remaining Edinburgh got themselves right back in it as fly-half Leonard dabbed through a delightful grubber for Scott to collect and fall over the line, but Laidlaw missed the conversion.
And Edinburgh were denied any points from the game as McFadden slotted a final penalty in the dying seconds to seal the win.
|Edinburgh Rugby Score Card|
|Leinster Rugby Score Card|