From the kickoff in Paris, Stade pinned their English visitors back. Peniasi Dakuwaqa scooted over untouched in the corner after three minutes.
Zack Henry converted. When he added a penalty after 12 minutes, Saracens had barely completed a pass or escaped their half.
The match changed after 21 minutes when flanker Sekou Macalou first tripped Ivan van Zyl as he picked the ball up from a scrum. Macalou then dived on the fallen scrum half, catching the South African's head with an arm. Referee Hollie Davidson opted for a red card.
Two minutes after the French international was sent off, Liam Williams, making his second Saracens debut after returning from a stint in Japan, swerved through the defence and slid over the line.
Fergus Burke converted and booted two penalties to give Sarries a 13-10 half-time lead.
Saracens again punished Stade for a card early in the second half. Stade lock Tanginoa Halaifonua caught winger Rotimi Segun in the head with his shoulder in a 43rd minute tackle. Halaifonua saw yellow. Segun had to be replaced.
Two minutes later, hooker Jamie George touched down after Saracens mauled forward from a lineout.
Saracens lost Rhys Carre for 10 minutes for leading with an elbow. After he returned, Burke booted his team to an 11-point lead in the 63rd minute.
Stade, a man short, rallied. Fullback Leo Barre looped round from a scrum to score and then converted.
But after the home team collapsed a maul in the 73rd minute and Hugo N'Diaye received a yellow, Saracens kicked to the corner and again drove over the line from the lineout.
Replacement hooker Theo Dan touched down and Burke converted to seal the first home loss suffered by a French club in this year's competition.
Saracens rose to second in Pool 3, a point behind Northampton.
In Italy, Bath started only four of the lineup that lost at home to La Rochelle in round one.
Treviso quickly pounced, racing into a 15-point lead after 15 minutes with tries from fullback Rhyno Smith and winger Louis Lynagh.
Hooker Niall Annett replied for Bath. But a breakaway try from an interception by Smith put the hosts 17-7 up at halftime.
Bath made a host of changes and fought back.
Flanker Josh Bailey touched down after 52 minutes. Prop Thomas du Toit, a starter last week and a half-time replacement on Sunday, added a try after 58 minutes. Orlando Bailey landed his third conversion to put Bath four points ahead.
Treviso were not done. They pressed furiously and after bungling a series of chances, won the game in the last minute when Bautista Bernasconi scored an unconverted try.
Treviso climbed to fourth in Pool 2. Bath are fifth, ahead of winless Bristol because of two losing bonus points.
