Cardiff are second in the URC and have three vital derbies over the festive period, which influenced coach Corniel van Zyl’s decision to rotate his squad after a win at Zebre.
Three players made debuts, with lively full-back Young starting while centre Elijah Evans and scrum-half Ieuan Davies came off the bench.
Cardiff were also boosted by the return of wing Stephens, and physical back Mason Grady continued his comeback on the other flank after starting in Parma.
On the down side, fly-half Ioan Lloyd was forced off in the first half by an injury to his right shoulder.
How Cardiff fare against Ulster at the Arms Park next Saturday is likely to shape their approach for rounds three and four in January given their URC play-off push.
Stade, fourth in the Top 14, were held up over the line twice before full-back Mathis Ibo opened the scoring in the 16th minute with a smart finish down the right after a line break by hooker Giacomo Nicotera.
The try-scorer then turned try-saver with a stunning effort to deny scrum-half Ellis Bevan, with Cardiff held up over the line.
Home tight-head prop Braxton Asi escaped without a card for a high tackle on James Botham, but the penalty led to Cardiff levelling in the 23rd minute.
They kicked to the 22, put the pressure on and worked the ball to the left for fit-again Stephens to cross from a terrific assist by Young.
The novice full-back, whose performance will have provided some cheer to Penarth RFC after their clubhouse was destroyed by fire last weekend, then went over himself in the same corner after 33 minutes when the forwards hammered away to create space out wide.
Cardiff led at half-time but trailed 17-10 with 50 minutes gone after flanker Tanginoa Halaifonua cantered through a gaping hole in midfield and then Ibo dived over for his second.
Stade had their bonus approaching the hour when scrum-half Thibaut Motassi took a quick tap and offloaded for replacement forward Juan Martin Scelzo to go over.
Further tries followed by Louis Foursans-Bourdette and wing Charles Laloi as Cardiff endured a point-less start to their campaign after being unable to add to Sebastian’s close-range finish.
