Their normal fluency and attacking invention never materialised at the Liberty Stadium, and defeat against Paris visitors Leicester next Saturday would leave the bell tolling on any quarter-final ambitions.
Stade scored a second-half try through wing Lucas Borges, yet full-back Juan-Martin Hernandez missed four kicks at goal from five attempts, while star backs Stephane Glas and Julian Arias were both forced off injured.
The Ospreys, despite being without injured British & Irish Lions Gavin Henson and Ryan Jones, made a mockery of their long quarter-final odds to progress from Pool Three.
Centre Sonny Parker pounced for a first-half touchdown, created by fly-half Shaun Connor's vision and pass, with Connor slotting a penalty and conversion before substitute Matthew Jones' 69th-minute penalty proved the clincher.
Stade had their share of high-profile absentees, as Agustin Pichot, Christophe Dominici, Brian Liebenberg, Sylvain Marconnet, David Auradou and Mauro Bergamasco all missed out, but there can be no excuses for such a poor display.
Stade were captained by prop Pieter de Villiers, while veteran fly-half Alain Penaud - who helped Brive lift European rugby's major club prize in 1997 - returned to the European Cup arena, aged 36.
Stade matched their billing as masters of champagne rugby by taking the field in pink jerseys, but it took the Parisians far longer than they would have imagined to break down a spirited Ospreys rearguard.
Hernandez hit the post with his opening penalty attempt, before Shane Williams hastily cleared to touch following a sublime Stade move featuring the handling skills of back-row forwards Shaun Sowerby and Remy Martin.
The Ospreys were in no mood to roll over, and their aggressive defensive work frustrated Stade, whose low-key start was underlined when Hernandez sent a 15-metre penalty chance wide.
The game was then held up for five minutes while Ospreys number eight Andy Lloyd received treatment following a clash of heads in midfield.
Lloyd was left prone on the turf, and referee Tony Spreadbury's prompt action in attending the player before medical staff arrived confirmed the injury's seriousness.
After a five-minute stoppage, Lloyd was stretchered off with his neck in a brace, and Glas also departed the action, although he was able to limp away.
The game then exploded into life with a flurry of scoring during first-half injury time.
Hernandez finally found his range to boot a penalty at the third attempt, yet the Ospreys stunned Stade through a 10-point burst in three minutes.
Connor was the architect of a superb Ospreys try, carving open Stade's midfield defence to send Parker over for a score that turned the game on its head.
Connor converted, and then added a penalty when Stade infringed in the shadow of their own posts, and Connor gratefully accepted a gift-wrapped three points.
And there was more good news for Ospreys fans during the half-time interval, when it was confirmed that Lloyd had regained consciousness, although he suffered a nasty facial gash following the alarming 39th-minute incident.
Hernandez missed another penalty four minutes after the restart, and a sorry tale of squandered chances continued for Stade when Arias could not ground possession after he decided to go it alone and ignore three unmarked team-mates outside him.
Arias suffered a knee injury in the process and was helped from the field, dealing Stade another major setback ahead of their crucial pool game at home to Tigers.
The Ospreys sensed a famous win, and with Stade's feathers well and truly ruffled, their forwards established base-camp inside the away 22.
Stade negated the threat, more by luck rather than judgement on occasions, and they looked to have clawed their way back into it when Borges crossed wide out, but Jones responded by landing an immediate penalty and the Ospreys had arrived on cloud nine.