Nick Evans produced a match-winning performance, kicking five penalties, two conversions and a sweet drop-goal in a faultless kicking display as Harlequins succeeded at keeping a spirited Wasps side at arm’s length.
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 New Zealand fly-half Evans’ golden boot guided Harlequins to victory.
Harlequins, who won this tournament in 2001 and 2004, will travel to 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.
Meanwhile, Stade Francais booked their place in the semi-finals with a 32-28 victory over rivals Montpellier at the Stade Charlety last night.
The game was a must-win for Michael Cheika’s side, who are out of touch with the Heineken Cup places in the Top 14 and relying on European success to add colour to another disappointing season. They now face Clermont Auvergne, who defeated La Rochelle in Thursday’s match.
The home side scored tries through Hugo Southwell, Tom Palmer and Antoine Burban while referee Alan Lewis awarded a penalty try.
Montpellier had surged ahead thanks to an early effort by Dimitri Pelo and the collective kicking exploits of Beneoit Paillaugue and Raphael Lagarde but faded in the second-half.
Harlequins 32 (Care, Fa’asavalu try each; N Evans 2 cons, 5 pens, drp gl) Wasps 22 (Jewell, Lindsay, Haughton try each; Walder 2 cons, drp gl);
Stade Francais 32 (Southwell, Palmer, Burban try each; Penalty try; Dupuy 2 cons; Beauxis con, 2 pens) Montpellier 28 (Pelo try; Paillaugue con, 3 pens; Lagarde 4 pens);
Sale 36 (Williams 2 tries; Addison try; Hodgson try, 2 cons, 4 pens) Gloucester 31 (Molenaar, May try each; Penalty try; Burn con, 3 pens; Taylor con, pen).