The decisive passage came in the second half as Leeds, having seen Martin Rodriguez kick nine straight points to get the visitors back into it, saw Lachie Mackay's pass strike Mauro Bergamasco on the head and fall loose.
Guillaume Bousses beat Semi Tadulala in a foot race for the ball and snatched the try, converted by Rodriguez to put Stade 16-10 up and complete a huge turn around.
That came after Leeds had dominated the early going.
Although Luther Burrell missed a penalty, Leeds got the first points of the game when Juan Gomez crossed the line.
Warren Fury missed the conversion, but after Stade had Benjamin Tardy sin-binned, Hendrie Fourie touched down one-handed to make it 10-0, although Fury missed again.
Rodriguez missed his first penalty as Stade struggled, but he made one just before half-time to get them on the board.
But the second half was a different story as a sloppy penalty for crossing at the restart allowed Rodriguez to cut the lead to 10-6 before another penalty in the 48th minute made it 10-9.
The lucky try made it 16 straight points for Stade before Ceiron Thomas' penalty cut it back to 16-13. However, a drop goal and a penalty from Rodriguez put the icing on the cake for the visitors.
A late miss from Thomas meant Leeds missed out on a losing bonus point.