A late drop-goal from Miah Nikora landed Connacht an epic 23-20 win over Bourgoin - and a spot in the semis of the Amlin Challenge Cup.
Replacement Nikora, the late hero against Edinburgh in the Magners League last week, showed his coolness under pressure once again, with the New Zealander kicking two penalties and the match-winning drop.
When scrum-half John Senio sniped through for Bourgoin's second try, converted by Benjamin Boyet, the French side led 20-14 and looked poised to defeat the number one seeds at the quarter-final stage for the second year running.
London Irish fell to them twelve months ago, but Connacht, aided by a yellow card for Boyet, raised their game superbly in the final 20 minutes, with their forwards doing the hard graft and Nikora nailing his kicks.
Penalties in the 66th and 73rd minutes levelled an absorbing tie and following a series of pick-and-goes from the home pack, the Kiwi out-half knocked over a well-struck drop goal to earn Connacht a gritty victory - their eighth on the trot at home.
Michael Bradley's men faced into a blustery wind in the opening half and Boyet had two early penalty attempts from the 45-metre range which he pushed to the right.
Connacht hit the front on the quarter-hour. The destructive George Naoupu bulldozed across the gain-line, Ian Keatley knocked a penalty to touch and experienced lock Michael Swift, with support from Brett Wilkinson, crashed over to the left of the posts.
Keatley landed the conversion but failed to punish France international Olivier Milloud for a scrum offence ten minutes later as his penalty effort drifted away from the target.
Connacht had their tails up but they missed out on a certain try in the 27th minute. Scrum-half Murphy got a favourable bounce as he chased his own kick but, with the Bourgoin rearguard completely stretched, lock Mike McCarthy knocked on with the line in his sights.
The hosts paid the price just three minutes later as Bourgoin shunned a kickable penalty and hooker Tone Kopelani was able to burst clean through the middle of a ruck and dive over in the right corner, with Boyet converting.
A foot in touch robbed Keatley of a try after good work by Niva Ta'auso and captain John Muldoon was also called back for a knock-on as Connacht pressed for their second try.
They did not have long to wait as, five minutes before the break, a wrap-around move involving Kiwi Naoupu in midfield put centre Keith Matthews roaring through a gap and he drew the cover to send Muldoon powering over for a converted score.
Boyet kicked Bourgoin back to 14-10 as half-time approached and although a late, pacy break from full-back Gavin Duffy engineered a penalty for Keatley, his drilled attempt from wide on the right faded in the difficult wind.
Bourgoin made changes for the second half, the most significant seeing Boyet reverting to full-back and Sebastien Laloo introduced. Indiscipline from Connacht soon allowed Boyet kick the first of two early penalty attempts, making it 14-13.
The wind-backed hosts struggled for territory and Bourgoin took a deserved lead in the 58th minute when Senio spotted a mismatch on the 22, darted through a gap and dotted down despite the best efforts of Duffy.
Boyet converted and the game seemed to be slipping from Connacht's grasp as Keatley missed a penalty from distance.
However, with Sean Cronin, Swift, Muldoon and Johnny O'Connor putting in heroic displays, Connacht rose to the challenge. As Murphy looked to pass left from a ruck in front of the Bourgoin posts, Boyet slapped the ball out of his hands and a yellow card was the only option.
Nikora, on for Keatley, landed the resulting penalty and, after Laloo had missed a drop goal, the Connacht replacement rewarded his forwards with the levelling penalty.
The momentum was clearly with Connacht now and, having missed a drop from further out just two minutes earlier, Nikora was right on the money with his 79th-minute attempt to send the westerners through to the last four of Challenge Cup for the first time since 2005.
Tries: Swift, Muldoon
Cons: Keatley 2
Pens: Nikora 2
Tries: Kopelani, Senio
Cons: Boyet 2
Pens: Boyet 2
Connacht: 15 Gavin Duffy, 14 Troy Nathan, 13 Niva Ta'auso, 12 Keith Matthews, 11 Fionn Carr, 10 Ian Keatley, 9 Frank Murphy, 8 George Naoupu, 7 Johnny O'Connor, 6 John Muldoon (capt), 5 Mike McCarthy, 4 Michael Swift, 3 Jamie Hagan, 2 Sean Cronin, 1 Brett Wilkinson.
Replacements: 16 Adrian Flavin, 17 Ronan Loughney, 18 Bernie Upton, 19 Mike McComish, 20 Conor O'Loughlin, 21 Miah Nikora, 22 Aidan Wynne, 23 Robbie Morris.
Bourgoin: 15 MaÃ«l Moinot, 14 Rudi Coetzee, 13 Piet van Zyl, 12 Alexandre Dumoulin, 11 Jean-FranÃ§ois Coux, 10 Benjamin Boyet, 9 John Senio, 8 Camille Levast, 7 Tim Cowley, 6 Julien Frier, 5 Coenie Basson, 4 B Williams, 3 Arnauld Tchougong Kamga, 2 Tone Kopelani, 1 Olivier Milloud.
Replacements: 16 Jean-Philippe Genevois, 17 Vincent Pelo, 18 Sylvian Charlet, 19 JÃ©rÃ©my Guillot, 20 Bogdan Leonte, 21 SÃ©bastien Laloo, 22 Matias Viazzo.
Referee: Andrew Small (England)
Assistant referees: James Jones (Wales), Robin Goodliffe (England)
Television match official: Graham Hughes (England)