Stade Francais secured a European Challenge Cup semifinal against Bath in Paris after grounding Ospreys at the Principality Stadium.
Despite having wing Josaia Raisuqe sent off for a second yellow-card offence just 12 minutes into the second half, Stade were good value for their 25-21 victory.
Prop Zurabi Zhvania, wing Julian Arias and substitute back-row forward Raphael Lakafia scored tries, while France international fly-half Jules Plisson added two penalties and two conversions to end Welsh interest in Europe this season.
The Ospreys delivered a Josh Matavesi try and late Tyler Ardron touchdown, plus three Dan Biggar penalties and a conversion, but ultimately they were architects of their own downfall, failing to make pressure count and gifting Stade opportunities in open play.
A 12,000-strong crowd -- the game was switched from Ospreys' usual Liberty Stadium home in Swansea due to Swansea City's Premier League game against Middlesbrough -- did its best to rally the home side but ultimately to no avail.
England head coach Eddie Jones was among the audience, casting an eye on Bath-bound Ospreys flanker Sam Underhill ahead of this summer's Argentina tour, yet Underhill could make little impression as Stade edged one step closer to the Challenge Cup final.
Biggar kicked the Ospreys into a 12th-minute lead and they went close to extending that advantage just two minutes later following a scintillating move started by scrum-half and skipper Rhys Webb.
Webb broke deep into Stade territory through trademark pace and power, and Ospreys looked to have enough attacking numbers out wide, but wing Keelan Giles's pass to an unmarked Biggar was ruled forward by referee Matt Carley after consultation with television match official Sean Davey.
It was a considerable let off for Stade, who continued to find themselves on the back foot, offering little in attack despite the promptings of their skipper, Sergio Parisse.
And they were their own worst enemy midway through the half when Raisuqe lashed out with his boot after being tackled by Giles.
Carley engaged in a prolonged discussion with the TMO but he decided that a yellow card was sufficient punishment and Stade cleared the danger to remain just three points behind after being outplayed in the opening quarter.
But the Ospreys were rocked by an eight-point Stade scoring burst in three minutes as Plisson tied things up through a penalty before the Ospreys were exposed in their wide defensive channel.
Stade attacked strongly and they had sufficient numbers out wide to allow Zhvania an unopposed run-in to put the French side ahead.
Biggar's second successful penalty then cut the gap but both he and Plisson missed chances just before halftime and Stade trooped off with an 8-6 interval advantage.
Plisson increased the lead to six points through an early second-half penalty but the Ospreys hit back immediately through a well-worked score that had Biggar at its attacking hub and ended with Matavesi diving over in the corner.
Biggar's touchline conversion attempt drifted wide but Stade were then reduced to 14 men for the remaining 28 minutes after Raisuqe received a second yellow card - this time for a technical offence - and an automatic red.
Biggar's resulting penalty edged Ospreys ahead but Stade were not fazed and they replied in style through a brilliant Plisson cross-kick that found Lakafia, who gathered on the bounce for an opportunist try.
Plisson converted effortlessly from the touchline, then Stade struck again, this time after Ospreys full-back Sam Davies' was intercepted on halfway and resulted in a try for Arias which Plisson converted for a 25-14 advantage.
Ardron crossed for a try with four minutes left, which Biggar converted, but it proved too little too late, and Stade marched on to the last four.
