The Swansea-based region fell four points short as the Parisians came out on top 21-25 at Principality Stadium on Sunday night.
Tandy reckoned his team had enough chances to secure victory, but paid the price for their mistakes.
He said: “I think we deserved to win the game, but ultimately we paid for some inaccuracies, not finishing off opportunities, and some ill-discipline in the first-half.
“We still had enough possession and opportunities to finish the game off.
“There is no hiding place, it is disappointing. One team has not done a lot to win the game, but that’s knock-out rugby.
“This week I don’t think it was an attitude thing, we were pretty inaccurate. We conceded two cheap tries and ultimately that is the game.”
Stade went down to 14-men after Josaia Raisuqe’s 52nd-minute dismissal for a second yellow-card offence.
But two second-half tries from Raphael Lakafia and Julien Arias combined with Zurabi Zhvania’s score before the break proved enough for the French side to book a semi-final clash with Bath Rugby.
Wales scrum-half and stand-in Ospreys captain Rhys Webb was left disappointed.
“It was very frustrating. It is very disappointing and one that got away ” Webb said.
“We felt in control of that game, we had a lot of possession, the accuracy and discipline cost us in the end.
“They scored two tries I do not think they should have scored. It was mistakes from us, silly mistakes. We had opportunities to score tries, but we cannot dwell on that now.”
Both Tandy and Webb believe it is vital now the Ospreys switch their attention to Leinster’s visit to the Liberty Stadium on Saturday.
Leo Cullen’s side toppled Wasps 32-17 at the weekend to seal a final four Champions Cup battle with Clermont Auvergne.
And Tandy reckons that fixture is perfect for his team to show their mettle.
He added: “We have Leinster on Saturday and there is no bigger or better game for us to be fired up for to get a performance and a result.
“It will be a big game and we want to get back to winning ways.”
Webb – who has 28 caps for Wales – echoed his coach’s thoughts.
“Leinster will be fully-loaded and confident after their win at the weekend,” he admitted.
“We need a big week of training and to get the little details right and look for a big performance.
“There is no bigger challenge than Leinster, but we have to pick ourselves up quickly and to finish off strong in the league.”
