Irish led 19-10 after 56 minutes, but fell away poorly in the second half to lose the Pool Two encounter.
The hosts made an unusual start in that they took the lead after six minutes despite having not had their hands on the ball.
After Edinburgh's Harry Leonard missed a penalty, the hosts went in front when, following an offence at the breakdown, full-back Tom Homer slotted over from 45 metres.
Homer struck again three minutes later, this time from just inside his own half.
Back came Edinburgh, hammering away at the Irish line, but referee Romain Poite let the hosts off the hook when he penalised Geoff Cross for an offence nobody else in the ground noticed.
Irish then drove Edinburgh off their scrum put-in to ease the pressure, only for Steve Shingler to hoof his clearance straight to Tim Visser, and back came Edinburgh.
Leonard moved the ball left, and Visser produced a clever pass for right wing Lee Jones, popping up on the left, to beat off a tackle by Matt Garvey and score by the flag. Poite sought advice from the video referee who was quick to award the score.
Leonard converted from a difficult angle to create a scoreline that better reflected the opening quarter.
Edinburgh conceded a soft penalty at a 23rd-minute ruck, gifting Irish another penalty and Homer's execution was again spot-on, regaining the lead for the hosts against the run of play.
Nick Kennedy soared to take an important lineout, thrown in by Edinburgh five metres from the Irish line, then England international Shontayne Hape produced a huge hit to halt David Denton in his tracks in midfield. Denton required treatment and was taken off looking groggy.
And it was Hape who nearly scored for Irish shortly after, hacking on a loose ball only to be just edged off touching down by Visser who matched the centre's pace.
That appeared to lift Irish who produced their best sequence of play on 32 minutes.
Adam Thompstone sparked it with a tackle- breaking surge up the left. Ross Sansom twice whipped passes left to right before Irish drove for the line. The initial attack was held up, but scrum-half Samson snatched up the ball and dived through the bodies for his first Irish try, Homer converting.
Edinburgh ended the half as they began, however, attacking Irish, and Leonard reduced the arrears 60 seconds from the break with a penalty.
Irish replied in kind shortly after half-time through Homer.
Leonard put a penalty wide before Edinburgh notched their second try on the hour mark.
Nick de Luca provided the slick pass that released Stuart McInally on a thundering run up the left and he went over the line. Greig Laidlaw kicked the conversion.
The game turned sour for Irish on 69 minutes. Poite sinbinned Hape for a shoulder charge on Sean Cox, and Laidlaw doubled the pain by edging Edinburgh in front with the resulting penalty.
Homer fired a penalty wide six minutes from time and Jebb Sinclair knocked on when Irish were on their final attack as Edinburgh held on.
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