But the Ospreys made a fist of it with tries by Sonny Parker and Lee Byrne keeping the home side under pressure, but two second-half penalty misses by Gavin Henson put the Welsh on the backfoot.
Stade opened the scoring when fly-half David Skrela landed a difficult penalty attempt from 40 yards out on the far touchline after 12 minutes.
But the Ospreys took the lead less than five minutes later when former All Blacks scrum-half dummied right from the scrum, leaving No 8 Ryan Jones to exploit the hole to his left.
Fly-half Gavin Henson worked a scissors with Sonny Parker and the big centre handed off a few would-be tackler to touch down for the try out wide.
Henson slotted the conversion as the Ospreys took the lead 7-3. This was not in the script as far as Stade were concerned and they punished their guests with a spell of intense pressure on the Ospreys' line.
Only a last-ditch tackle from flanker Steve Tandy and some good covering defence by Marshall kept the hosts at bay.
The pressure told when Tandy pulled a maul down and from Skrela kicked the resulting penalty to narrow the lead to 7-6.
Use of the boot saw the Ospreys reduced to 14 men when former All Black flanker Filo Tiatia was yellow carded and Skrela landed his third penalty to wrestle the lead back at 9-7.
The Ospreys went on the attack with Henson twice going wide with drop goal attempts and inside-centre Andrew Bishop was hauled down just a yard short of the line.
The home side failed to capitalise on their numerical advantage to make it 9-7 at the break.
The Ospreys brought on Wales forwards Jonathan Thomas and promising young talent Alun Wyn Jones.
They spent the opening exchanges of the second-half in Stade territory and Henson could count himself unlucky to see his penalty attempt hit the post.
Marshall was brought down on his way to the try line, but the home defence killed the ball presenting Henson with another penalty attempt which went wide.
Shortly after, James Hook took over at fly-half for luckless Henson.
Ryan Jones found himself offside at a ruck to allow replacement fly-half Lionel Beauxis to put his side 12-7 with a well-struck penalty on 18 minutes.
The sucker punch was landed just seconds later when a flowing move cut the Ospreys defence to pieces and ended with Beauxis giving a scoring pass to Italian centre Mirco Bergamasco to dive in under the posts.
Beauxis' conversion put Stade in the driving seat at 19-7, before a try by replacement forward Rodrigo Roncero killed off the Ospreys.
Lee Byrne's late try was no more than consolation before No 8 Sergio Parisse wrapped up the scoring with a late drop goal.