Centre Cory Allen was the man to send them on their way with a brace of tries in the opening half and fly-half Sam Davies was on target with the conversions and three penalties.
Stade Francais had to be content with a single try from winger Jimmy Yobo to add to two penalties and a conversion from South African fly-half Morne Steyn as they trailed 23-13 at the break.
All hopes for the visitors seemed to disappear when centre Mailetoa Hingano was sin-binned for killing the ball and Ospreys quickly took advantage when replacement prop Rhodri Jones showed an impressive burst of pace to score a breakaway try.
The visitors were reduced to 14-men for a second time when Stephane Clement was yellow carded and Ospreys finished off strongly with additional scores from winger Luke Morgan, an interception score from replacement centre Owen Watkin and a penalty try. Further conversion followed from Sam Davies and James Hook.
Stade Francais, whose only second half points came from a try by winger Julien Arias converted by Steyn, will look to make a better impression when they host Ospreys on Friday night.