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.

Rio the red kite was found near Leitrim, County Down, with an infection in her foot.
It meant the bird could not flex her talons, which affected her ability to feed.
She became weak and unable to fly.
She was reintroduced to the wild after being treated by a vet.
Rio was one of 80 red kites introduced to Northern Ireland and has raised five chicks.
She is important to the programme because she has bred consistently since her release.
The red kite has been under pressure in south Down since its reintroduction in the years between 2008 and 2010.
Three birds have been shot, a further 12 have been poisoned and 12 more have been found dead with no clear cause.
There are just 12 breeding pairs, well short of the 50 pairs experts say are required for a sustainable population.
Claire Barnett is a senior conservation officer with the Royal Society for the Protection of Birds (RSPB).
She said they have tried hard to get the message across that the birds do not pose a threat to livestock - especially lambs.
"The bird actually has small, weak feet so there's no way they're going to be able to pick up anything bigger than a small rabbit.
"Their main diet is made up is things like mice and worms and rats and road kill," Ms Barnett said.
There is some good news though - the breeding pairs have produced 71 chicks and RSPB staff said there is evidence that the birds have begun to spread out.
They are asking the public to send them details of any sightings.
Red kites are quite distinctive. They have a five and a half foot wingspan and a deeply forked tail.
Last season's beaten finalists Harlequins kicked off the European Challenge Cup with a dominant home victory against Stade Francais.
[NEXT_CONCEPT]
A key member of a breeding programme, designed to reintroduce an impressive bird of prey in Northern Ireland, has been returned to the wild after being found injured in a field.