Tom Varndell’s 72nd-minute try, which was converted by out-half Andy Goode, sent Tigers through to a last-four appointment with Welsh challengers Llanelli Scarlets at Walkers Stadium in three weeks’ time.
Leicester’s victory also means England will have a record three clubs in the semi-finals after Northampton shocked Biarritz and booked a clash against 2004 champions Wasps.
But they left it late to send Stade spinning out of Europe, taking charge just when it looked as though fly-half David Skrela – who slotted four penalties and a drop-goal – would steer the Parisians through.
Leicester boss Pat Howard insisted: “The treble is our goal. It is a very hard goal, but we believe we have a squad that can attempt it. We want to fight on all fronts, and you have got to aim high in whatever competition you take part in.”
Howard also paid tribute to Leicester skipper Martin Corry, who defied an elbow injury sustained last weekend to produce a towering performance before going off midway through the second half.
Howard added: “The medical staff did a great job with Martin. He has got a massive heart, and his presence makes a huge difference.”
Corry added: “I think I aged quite a few years at the end. It was one of those tough ones that you couldn’t really bear to watch.
“It was tremendous guts from the players to hold on. We made some mistakes, but in terms of determination and courage, it was 10 out of 10 from our players.”
Stade coach Fabien Galthie offered no excuses for his team’s exit. “Credit to Leicester for their performance, but it is very disappointing for Stade Francais and for the French clubs in general that there are no representatives in the semi-finals.”
Full-back Juan Martin Hernandez, the game’s outstanding back, scored a first-half try, but Varndell came up trumps when it mattered following an early touchdown by Seru Rabeni. Goode kicked 11 points.
Stade were without injured forwards Sylvain Marconnet, Dimitri Szarzewski and David Auradou, while Italy flanker Mauro Bergamasco missed the sold-out tie through suspension.
Leicester scorers — Tries, Rabeni, Varndell. Cons, Goode. Pens, Goode 3.
Stade Francais scorers — Tries: Martin Hernandez. Pens: Skrela 4. Drop Goals: Skrela.
G. Murphy, Rabeni, Hipkiss, Gibson, A. Tuilagi, Goode, Ellis, Ayerza, Chuter, White, L. Deacon, Kay, Moody, Jennings, Corry.
Replacements: Varndell for Rabeni (54), Cullen for L. Deacon (64), B. Deacon for Corry (68).
Martin Hernandez, Arias, Glas, Liebenberg, Dominici, Skrela, Pichot, Roncero, Kayser, De Villiers, du Plooy, M. James, Rabadan, Martin, Parisse.
Replacements: M. Bergamasco for Glas (68), Blin for Kayser (57), Marchois for du Plooy (60), Burban for Martin (51).
Alan Lewis (Ireland).