Care's moment of magic - a sublime reverse pass inches from the touchline - teed up Matt Hopper's try which gave Quins a 13-3 lead at half-time and kept their hopes of reaching the knockout stages alive.
Clermont hit back soon after the break when a bouncing ball evaded England full-back Mike Brown and Fijian flyer Napolioni Nalaga darted in before the Top 14 leaders levelled at 13-13 when Sitiveni Sivivatu went over in the corner.
A late Brock James penalty then handed Clermont the victory and ensured Harlequins, who lost Care to a twisted ankle on 49 minutes, cannot reach the quarter-finals.
England begin their Six Nations campaign away to France on February 1 but O'Shea remained hopeful Care will recover in time.
"Danny twisted his ankle and he's got a protective boot on," said O'Shea. "We'll review it on Monday and see where he's at. It's pretty sore at the moment but hopefully he will be alright.
"I hope he'll be okay for a few weeks' time but I doubt he'll be all right for next week.
"We didn't start particularly well but then a piece of absolute genius from Danny changes the energy our way and we were pretty much on top for the rest of the first half.
"Danny's piece of skill was quite exceptional. You can't practice that. People may ask if we overplayed in the second half but it was Danny tapping and going 35 yards from our own line. He gave us a bit of momentum and he's a game changer.
"Mike Brown had covered the ball and it bounced badly. You can't blame anyone when that happens."
Victory saw Clermont - last year's beaten finalists -become the first team to reach the last eight but head coach Vern Cotter believes his side were far from their best.
"We didn't have a particularly good first half. We created opportunities but we didn't score from them," he said.
"We changed tactics slightly at half-time to try and play a little bit more in their half. We knew it may mean we lose the ball more often but we backed our defence and just looked for opportunities.
"We got one [try] from a little kick in behind and we got another one from a couple of good passes out wide.
"We didn't get any joy from our kicks but the character was there to keep slugging it out. We weren't perfect in what we tried to do, we lost a lot of ball, but we managed to do enough to win which got us the qualification we wanted.
"We managed to put them in a situation where they needed to play to get back into the game and just backed our defence."