With the clock showing eighty-five minutes the twenty-two year old replacement outside half was given the task of shooting for goal from the halfway line to try to draw the match.
Heineken man-of-the-match, Ryan Jones, had given away a penalty for handling at a ruck and his calculated gamble backfired as Beauxis hit the target to make it 22-22 with thirty seconds left to play.
Having clinched a losing bonus point at the death at Sale in Round Four, Beauxis's penalty put the Parisians in control of Pool Three and on course for a quarter-final berth.
The visitors, who had lost by a point at the Liberty Stadium the previous season, totally dominated the first half and were good value for their 16-6 lead, which included a try from full back Juan-Martin Hernandez.
Beauxis came on to replace the injured David Skrela four minutes into the second half extended the French side's lead with a thirty metre penalty that made it 19-6 with twenty-seven minutes to play. Then the game was turned on its head by the sin binning of Stade's South African lock Boela du Plooy for killing the ball in front of his posts.
By the time he returned to the field, James Hook had kicked two penalties and converted a Nikki Walker try and the scores were level at 19-19.
The Ospreys revival continued when Hook hit the mark from forty metres in the seventy-third minute and it looked as though the Welsh region were going to hang on for another famous victory. There were a number of twists and turns in the closing thirteen minutes before Chris White blew for the end of the game in the sixth minute of injury time.
The Ospreys' replacement scrum half Jason Spice picked up a yellow card after he had been on the pitch for only three minutes and then Beauxis conceded a penalty on the home line for not releasing, it looked as if the visitors were bound to score. Then came that eighty-fifth minute penalty and a tournament saving kick from the Stade Francais youngster.