Remi Bonfils, Hugo Bonneval and Geoffrey Doumayrou, plus a penalty try, put the home side on course to avenge last week's defeat at Murrayfield.
Edinburgh's only first-half reply had been a Jason Tovey penalty at 7-0 down.
But tries from Fraser McKenzie, Viliame Mata and Sam Hidalgo-Clyne brought the Scots to within six points.
And the visitors were left to rue a poor first-half and two conversion attempts that struck a post.
Edinburgh remain top of their qualifying group, but Harlequins will overtake them should they win their game in hand on Saturday at home to bottom side Timisoara Sarcens of Romania.
Stade Francais applied early pressure and it paid off when hooker Bonfils went over for the opening try, with Jules Plisson converting to establish the lead after 12 minutes.
Ben Toolis thought he had replied for Edinburgh, but the TMO deemed it no try for a double movement on the lock's approach to the line.
Tovey did reduce the arrears with a penalty, but Bonneval's try edged Stade Francais further in front.
More home pressure told with a penalty try, converted by Plisson, and a costly yellow card to Edinburgh prop Simon Berghan for the scrum infringement.
Stade Francais were now well in control and it was followed, within five minutes, by another try, this time by Doumayrou, and another Plisson conversion.
It was the home side's turn to have someone in the sin bin following Sylvain Nicolas' ruck infringement three minutes before the break.
McKenzie made the man advantage tell shortly after half-time, forcing his way over from close range, but Tovey's conversion attempt rebounded off the upright.
Edinburgh had fought back from 20-3 down at half-time to win 28-23 at home and, when Plisson crossed the line but had a try ruled out by the TMO for an infringement by Sekou Macalou, they sensed their chance again.
Blair Kinghorn crossed the line for Edinburgh only to have the ball knocked out of his hands as he was about to score.
Mata did slide over to reduce the deficit, with Tovey successful on this occasion.
Hidalgo-Clyne gave Edinburgh hope of victory with a try with two minutes left on the lock, but Duncan Weir's conversion hit a post and time ran out for the visitors.
Stade Francais: Bonneval, Sinzelle, Doumayrou, Danty, Raisuge, Plisson, Daguin, Felsina, Bonfils, Slimani, Pyle, Pape, Ross, Macalou, Parisse, Nicolas.
Replacements: Arias for Doumayrou (58), Steyn for Plisson (64), Zhvania for Felsina (58), Burden for Bonfils (66), Melikidze for Slimani (58), Nicolas for Ross (35), De Giovanni for Nicolas (59). Not Used: Coville.
Sin Bin: Nicolas (38).
Edinburgh: Kinghorn, Hoyland, Allen, Dean, Brown, Tovey, Hidalgo-Clyne, Dickinson, Ford, Berghan, McKenzie, B. Toolis, Du Preez, Watson, Manu.
Replacements: G. Bryce for Allen (27), Weir for Brown (49), Dell for Dickinson (58), Cochrane for Ford (58), Carmichael for McKenzie (59), Mata for Manu (59). Not Used: McCallum, Kennedy.
Sin Bin: Berghan (28).
Referee: Dudley Phillips (Ireland).

The 30-year-old joined the Blades in the summer but made just one appearance, before being loaned out to former club Walsall last month.
He played seven times for the Saddlers and boss Dean Smith had been keen to retain his services.
Butler joins a Doncaster side that is currently one point above the relegation zone.
Edinburgh suffered their first defeat in European Challenge Cup Pool 5 despite a spirited second-half fightback at Stade Francais.
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Doncaster Rovers have signed Sheffield United defender Andy Butler on loan until January.