Quins overcame the loss of lock George Merrick to injury to score five first-half tries and secure a bonus point, with Stade managing a solitary reply.
Tim Visser's intercept try came as Stade rallied early in the second half.
Clement Daguin and Willem Alberts tries gave Stade faint hope that Marland Yarde's late try quashed emphatically.
It was an impressively dynamic first-half performance in front of watching England boss Eddie Jones, while it was their defensive mettle under more intense pressure in the second period that would have stood out.
Quick shifts and sharp set-piece work set up Joe Marler's try from a rolling maul, and after Merrick's injury stoppage, a two-try blast from Joe Marchant and Visser established control.
Sekou Macalou returned a skewed Yarde kick for Stade's opener, but two bludgeoning James Chisholm scores extinguished any momentum the Top 14 side had before the break.
Gonzalo Quesada's side threatened to breach a creaking Quins defence in the second half before Visser pounced to increase Quins' buffer.
The visitors did find some consolation with scores from Daguin and Alberts but they came too late to influence the result, particularly when Yarde's score took Quins beyond 40 points.
Harlequins: Brown; Yarde, Marchant, Roberts, Visser; Swiel, Dickson; Marler, Buchanan, Collier, Merrick, Horwill (capt), Robshaw, Wallace, Chisholm.
Replacements: Gray, Evans, Jones, Twomey, Luamanu, Mulchrone, Jackson, Marchant.
Stade Francais: Camara; Vuidarvuwalu, Bosman, Williams, Sinzelle; Plisson, Daguin; Taulafo, Burden, Slimani, Pyle, Pape (capt), Macalou, Lakafia, Parisse.
Replacements: Bonfils, Zhvania, Melikidze, De Giovanni, Alberts, Coville, Fartass, Millet.
Referee: Ben Whitehouse.
Last season's beaten finalists Harlequins kicked off the European Challenge Cup with a dominant home victory against Stade Francais.