Vuidravuwalu scored two of Stade's four tries and helped lay on the other two for full-back Hugo Bonneval, with fly-half Jules Plisson, scrum-half Jerome Fillol and replacement Jerome Porical all contributing points through the boot.
Bath secured a penalty try and consolation scores from replacements Michael Claassens and Jack Cuthbert, while Stephen Donald kicked five points.
After defeats for Gloucester and Wasps, Bath were the last English hope in the second-tier tournament. But they were ruthlessly exposed by a Stade side who, despite languishing in the lower reaches of the Top 14, played with a fluency that was simply beyond their hosts.
In addition to Vuidravuwalu, there was another Herculean display by captain Sergio Parisse to marvel at, while Bonneval was a classy presence throughout as Stade advanced to a last-four meeting with Perpignan.
There had been positive signs for Bath during the opening exchanges as Kyle Eastmond, with a chance to impress Lions coach Warren Gatland, ducked under two tackles in midfield, but he was brought to ground by Bonneval as he tried to step his way to the line.
It was the French side who registered the first points through a Plisson penalty, and they followed it with the first of Vuidravuwalu's brace.
The ball squirted out the side of a Bath ruck and crisp hands along the width of the field put the wing in at the corner, with Fillol converting.
Full-back Bonneval then sparked a thrilling Stade counter-attack which would have resulted in another try if not for a crucial tackle from Tom Biggs.
Fillol and Donald exchanged penalties before the Bath fly-half, who it was announced this week will leave the club at the season's end, gifted Stade their second try.
His wild pass was picked up by Geoffrey Doumayrou who flipped the ball beautifully to Jeremy Sinzelle, who in turn sent Vuidravuwalu on a 60-yard dash to the line.
Fillol again converted to make it 20-3. Stade were in complete command and even managed to get through the 36th minute sin-binning of prop Rabah Slimani, for taking Nick Abendanon out in the air, without conceding a point.
But Bath gave themselves hope upon the prop's return, worming their way to the line with a controlled rolling maul which Stade desperately tried to collapse.
Referee Nigel Owens had no hesitation in awarding a penalty try and dispatching lock Scott Lavalla to the bin.
Even with their numerical disadvantage, Stade looked more threatening than their sluggish hosts and only a superb last-ditch tackle from Francois Louw stopped Plisson jinking over.
It was a brief reprieve as Stade worked two wonderful tries for Bonneval in two minutes.
Vuidravuwalu was the provider for the first as he cantered past four Bath defenders to give his full-back an easy finish.
The second was a superb sweeping length of the field move. Vuidravuwalu broke down the right and linked with Doumayrou and Julien Arias to send Bonneval in again. The score left Bath 20 points adrift with 20 minutes to play.
They got a try back as Claassens dotted down from another purposeful maul, only for Plisson to respond by landing a huge drop goal from halfway with a minimum of effort.
Cuthbert got a late consolation for the hosts but Stade deservedly had the last word as a Porical penalty dropped over after hitting the crossbar.