The Aviva Premiership side thought that they had taken the lead after 14 minutes when Peter Betham crossed the line, but the try was ruled out after a blocking foul in the build-up.
Stade took the lead on the half-hour mark when Julien Tomas crossed the line after pressure on the Leicester try line.
The conversion was missed but Morne Steyn made up for the error three minutes after by placing a kick through the posts.
Leicester took a 12-8 lead into half time when Matt Smith finished in the corner.
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Brendon O'Connor scored a debut try from the restart and Owen Williams added a penalty kick shortly after as Leicester stretched their lead.
Owen Williams added another penalty but a Stade try took the score to 30-20 when Julien Arias's try was confirmed by the TMO.
Seremaia Bai added three more for Leicester in the final minutes to confirm the victory for the English side.