The newly-promoted side's second win of the season came courtesy of tries by Rida Jaouher and Lucas Dupont, with Valentin Courrent kicking 16 points.
It was Stade's second loss in the opening three games of the season.
The tries both came in the second half after Grenoble went in at the break 9-6 up. Stade managed to pull level from Jerome Porical's boot but Jaouher crossed the line soon after to put the hosts back into the lead.
Grenonble finished strongly with Dupont touching down to seal a memorable win.