Ospreys back-line coach Gruff Rees conceded their Heineken Cup hopes are over after they suffered 15-9 defeat at Castres.
Dan Biggar slotted three penalties at Stade Pierre Antoine but Rory Kockett and replacement Geoffrey Palis kicked the Top 14 champions to victory.
Castres saw Palis and Seremaia Bai sent to the sin-bin but managed to keep the Ospreys at bay as they went in search of the win.
The loss means the Ospreys are all but mathematically out of Europe at the group stages for the fourth season in a row, following earlier defeats to Leinster and Northampton.
"Effectively we are out of the competition," said Rees.
"But it was a huge effort by all the boys, from Alun Wyn Jones to young Daniel Suter - they all gave it everything.
"We have had a very difficult week with only one full training day but the boys applied themselves and gave everything they could for the shirt.
"There were opportunities at the end to win but it didn't happen and we have to look forward to the next game."
The victory moved Castres into second spot on the Pool One table. They are level on points with pacesetters Leinster but have played a game more. Leinster face Northampton at Franklin's Gardens on Saturday.
Castres lock Richie Gray admitted the Ospreys frustrated his sidebut he was delighted to secure the win.
"The Ospreys made life difficult for us at the breakdown and defended very well," he said.
"But we expected that, they are a very competitive side.
"We were down to 13 men at the end but we managed to hold out and I think that shows how hard this team works for each other.
"We did so well to hold out in that period. It was a real team effort.
"The win puts us in a good position in the group. Now we have a very difficult trip to the Ospreys next week but it is one we feel we can do well in."