Wales’ Capital Region lost Tom James before the game even kicked off and then saw Rhun Williams and Rey Lee-Lo depart early in the first-half.
They were undeterred and led 14-10 at the break after Cook and Williams finished some fine attacking spells in the space of three minutes.
But the pressure from star-studded Stade told in the second-half as Danny Wilson’s depleted side were hampered by further injuries.
They lost Steve Shingler, Willis Halaholo, Matthew Rees and Kirby Myhill and played the final quarter with Ellis Jenkins in the centre and Tomos Williams on the wing.
Stade added five tries in the dominant second-half, while Cook’s second was merely a consolation despite its spectacular nature.
The Blues were dealt a blow ahead of kick-off after Tom James pulled up in the warm-up and that was further compounded when his replacement, Rhun Williams, turned his ankle in the opening minutes.
Morné Steyn opened the scoring with a simple penalty on nine minutes, following a high arm in the tackle, and the bad luck continued when Rey Lee-Lo hobbled off.
However, the setbacks did not deter Danny Wilson’s side and they displayed plenty of attacking intent in perfect conditions.
They almost claimed the opening break after Willis Halaholo sliced through the Stade defence and found Lloyd Williams in support.
He combined with Cuthbert, but the wing, celebrating his 100th appearances, was not quite able to round the last man. The ball was kept in play by Williams, who found Navidi but he was bundled into touch and the opportunity lost.
But when Stade overthrew the ensuing lineout they had a second chance and grabbed it with both hands. They went through a succession of phases and a beautifully delayed Anscombe pass put Cook through the hole.
Anscombe converted to give his side a 7-3 lead and they continued to attack at every opportunity, moving Stade’s big pack around the pitch.
Following a quick lineout inside their own half, the ball was spread left to right to Cuthbert. He raced around his opposite number Sekou Macalou and drew the last defender.
It appeared as though he had put Willis Halaholo clear but the centre was unaware of the covering defence and was caught short.
They hammered away at the French hosts, with the try-line at their mercy, and eventually number eight Williams crashed over. Anscombe added the extras to make it 14-3.
Twice Stade responded and broke clear but some heroic last-gasp defence kept them at bay. That was until the 31st minute when a superb sleight of hand from the ever-impressive Sergio Parisse put Waisea Nayecalevu away.
Steyn converted to cut Cardiff Blues’ lead to just four points and they continued to press in the closing stages of the half.
Stade sensed blood and turned down three shots at goal, instead kicking to the corner, but on each occasion the Blues defended the driving lineout magnificently.
They eventually earned a penalty of their own to clear and the score remained 14-10 at the interval.
The Blues started the second-half well but soon found themselves trailing after a turnover in midfield. Antoine Burban made the initial inroads and following another touch of class from Parisse, Macalou surged down the left.
Cuthbert made the try-saving tackle but the Parisians maintained possession and Giorgi Melikidze soon powered over.
Steyn maintained his 100 per cent strike rate to open up a three-point advantage and they stretched that further when a further period of pressure saw Mathieu De Giovanni cross the whitewash.
Injuries continued to hamper the Blues with Shingler, who had bravely battled on with a lower leg issue, eventually forced off on 50 minutes.
That led to yet another reshuffle of the backline with Garyn Smith moving into his more familiar position of centre and Tomos Williams entering the fray as a makeshift wing.
Stade took a major stride toward victory when Macalou claimed their fourth try and the Blues appeared to be staring down the barrel but they responded with aplomb.
After a talking to under the sticks they rediscovered their attacking venom and quickly hit-back with Maceuley Cook claiming a second following an Anscombe cross-kick, which was kept alive by an acrobatic Tomos Williams catch-pass.
Anscombe converted but any hope of a comeback was dashed as Macalou completed his brace and Laurent Panis finished a scintillating counter-attack from Stade’s own line.
With full time approaching, Plisson's cross-field found Macalou in acres of space on the left, launching another counter-attack, which was finished by replacement Daguin to cap off the victory for the home side.