The French visitors got a late try from Italian international Mirco Bergamasco, but it was mere consolation.
Julien Saubade crossed with 10 minutes left and Mirco Bergamasco's late 70m run-in gave Stade a bonus point.
The Italians took the early initiative and Mirco Bergamasco slotted two penalties before Dan Parks replied to square the scores at half-time.
Britain could be celebrating two class wins on Sunday, with Darlington's Guy Wilks in second, just 45secs behind junior class leader Mirco Baldacci.
Italy claimed their solitary try as Mirco Bergmasco finished off a sweeping move after 72 minutes to cap a mixed day for the Bergamasco brothers.
Guy Wilks, benefiting from Suzuki team-mate Mirco Baldacci rolling his car into retirement on Sunday morning, won the junior class and took the championship lead.
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They came desperately close to scoring through Rougerie just before the hour mark, only for him to be denied by a superb Mirco Bergamasco tackle.
Ellis scored England's fourth try after the break and although Italy cut the gap through Mirco Bergamasco, Mark Cueto had the final word for the hosts.
Lachie Turner's score established an 8-3 lead for Australia but a drop goal and Mirco Bergamasco's try put the hosts in the driving seat at 14-11.
By now, Mirco Bergamasco had been handed the kicking duties for Italy and while he missed the conversion, he did notch a penalty five minutes into the second half.
Mirco Bergamasco sealed the bonus point for Stade by running in from long range after more turnover ball, while Barry grounded a consolation try for Quins in injury time.
Italy's kickers Mirco Bergamasco and replacement fly-half Luciano Orquera missed 10 kickable points between them and Italy, who conceded five penalties, spurned some kickable opportunities in the second half.
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