Bourgoin (6) 21
Lee Byrne scores the Ospreys second try
Tries: Wihongi, Denos
Pens: Laloo 2, Parra
Ospreys (18) 28
Tries: Williams, Byrne, Thomas
Cons: Hook 2
Pens: Hook 3
The Ospreys secured a place in the Heineken Cup quarter-finals for the first time after seeing off Bourgoin.
James Hook and Sebastien Laloo exchanged early kicks but tries from Shane Williams and Lee Byrne gave the Welsh side the advantage.
Bourgoin refused to give up and Karena Wihongi scored in the second half.
Ospreys lost Byrne in the dying moments to a yellow card but Jonathan Thomas eased the pressure before Florian Denos made it a nervy finish with a try.
Bourgoin took the lead after three minutes after Ospreys penalised for killing the ball.
Sebastien Laloo made no mistake with his kick, but the hosts lead lasted just two minutes as James Hook levelled with his own kick.
Moments later, Laloo restored advantage to make it 6-3 after Hook failed to roll away.
Bourgoin looked comfortable in defence as the Ospreys started to add pressure and gave away a penalty for offside, enabling Hook to level again midway through the first half.
The visitors were given a couple of let-offs almost immediately when Laloo missed his third attempt after Tiatia flew in with a high tackle.
And soon after they almost conceded the first try when Chris Wyatt burst down the right, only to see his inside pass cut out by Ryan Jones.
And that was costly for Bourgoin as the Ospreys took the advantage through Shane Williams getting his 14th European try.
Adam Jones passed out to Hook who found Williams in space to go over. Hook made it 13-6 with his conversion.
It didn't take long to extend that lead as they pushed forward and Henson laid off to Lee Byrne who dummied to cross in the corner.
An unsuccessful conversion kept it a 12-point lead.
The Ospreys were made to defend as Bourgoin tried to push forward, but struggled to find a way through in the closing stages of the first half.
Sonny Parker thought he had given the Welsh region the perfect start to the second half, but his try was ruled out for crossing in midfield.
And then it was Bourgoin's turn to feel hard done by when Matthieu Nicolas's try was disallowed for a forward pass by Morgan Parra.
But the home side did get the try they deserved when the video referee Geoff Warren awarded replacement Karena Wihongi's drive over.
They could not make the most of it though as Laloo missed the conversion.
The Ospreys were enjoying much of the possession, but struggled to make any real impact, in what was not a vintage performance.
But with nine minutes to go, Hook gave the Ospreys the breathing space they needed kicking a penalty from 35m in front to the post to lead 21-11.
Parra reduced the deficit in the closing stages and Byrne was then sin-binned for an illegal challenge on Mickael Forest making it a nervous finish for the Ospreys.
As the hosts pushed for the line, Henson intercepted and broke, laying off to Williams who freed Jonathan Thomas to go over, with Hook converting.
Denos pulled a try back with Parra converting, but it was too late for Bourgoin as Ospreys booked a trip to Saracens.
Ospreys coach Lyn Jones:
"We were right up for it today. I am very pleased for the players, they deserve every plaudid they get.
"When you are chasing something, there is a level of desire deep down and it drove us today through a lot of pain."
Bourgoin: A. Forest, M. Nicolas, Coetzee, Laloo, Coux, Parra, M. Forest, Tchougong, Genevois, Cardinali, Pierre, Wyatt, Frier, Petrilli, Jooste.
Replacements: Viazzo for Laloo (66), Denos for Coux (57), Sourgens for Cardinali (63), Monzeglio for Wyatt (57), Williams for Petrilli (57).
Not Used: Wihongi, Carmona.
Ospreys: Byrne, Walker, Parker, Henson, S. Williams, Hook, Marshall, D. Jones, Bennett, A. Jones, Gough, AW. Jones, R. Jones, Holah, Tiatia.
Replacements: James for D. Jones (72), Evans for Wyn-Jones (74), J. Thomas for Tiatia (66).
Not Used: Shervington, Phillips, Taylor, Vaughton.
Sin Bin: Byrne (80).
Ref: Wayne Barnes (England).