New Zealand fly-half Nick Evans produced a match-winning performance for
Harlequins had roared into a 14-0 lead with tries from Danny Care and Maurie Fa'asavalu before some of the crowd had even taken their seats.
Wasps responded well with tries from Seb Jewell, Tom Lindsay and Richard Haughton but Evans's golden boot guided Harlequins to victory.
Harlequins, who won this tournament in 2001 and 2004, will face either Munster or Brive in the semi-finals.
For Wasps, defeat effectively signals the end of their season. Winning the Amlin Challenge Cup was their only realistic route back into Heineken Cup rugby next season.
A try from former rugby league man Dimitri Pelo and 11 points from the boot of scrum-half Benoit Paillaugue put the visitors into a 16-0 lead after just 22 minutes.
But tries from Hugo Southwell and England lock Tom Palmer brought Stade back into the game and they trailed 19-12 at the break.
A penalty try and a score from replacement Antoine Burban plus a conversion and two penalties from Lionel Beauxis sealed the win, three penalties from Raphael Lagarde proving not enough for Montpellier.
Stade will now face Clermont Auvergne in the last four.