The Parisiens have now taken maximum points from their opening two games of the tournament to keep the pressure on Gloucester, who play Ulster today, and condemn Cardiff yet more European misery.
During a cautious first-half it had looked promising for the Blues, who have now lost 13 of their last 15 Heineken games, until young wing Julien Arias lit up the game with a stunning 70-metre solo effort.
Arias burst onto Augustin Pichot's pass, handed off two tacklers before leaving full-back Craig Morgan on his backside and red-faced to score in injury time.
His team 11-3 behind, Morgan responded in the best possible way and fashioned a try from a gutsy chip over the top in midfield just three minutes into the second half.
The bounce was kind and once he had shipped the ball out to Williams, the try for wing Dean Dewdney was a formality despite the best efforts of Christophe Dominici.
But just as the Blues had hauled themselves back into the game, Craig Quinnell's indiscipline undermined all that hard work. The big second-row was shown the yellow card for repeatedly straying off-side and the loss was immense.
The Blues had already virtually surrendered the lineout but with one less defender, their pack struggled to hold back a clinical Stade pack who smelled victory. Against seven men, the French champions made light work with two almost identical tries from driving lineouts through Mauro Bergamasco and Sylvain Marconnet as Quinnell filled a large space in the sin bin.
A second Dewdney try, crafted by Rhys Williams, offered a glimmer of hope until Gonzalo Quesada charged down the Cardiff centre's kick for the try that bagged a bonus point.