Leicester and Saracens joined Aviva Premiership champions Harlequins in the last eight, with only Tigers facing an away tie. Richard Cockerill's men progressed as sixth seeds after knocking out four-time European champions Toulouse, which means a trip to Jonny Wilkinson's Toulon in early April.
Saracens will be at home to last season's runners-up Ulster with Quins hosting Munster, whose bonus-point victory over Racing Metro sent holders Leinster crashing out, while the last eight is completed by an all-French affair between tournament favourites Clermont Auvergne and Montpellier.
The semi-final draw, meanwhile, sees Clermont or Montpellier drawn to play in their own country, but not their normal home ground, against Quins or Munster, with Saracens or Ulster hosting Toulon or Leicester.
Three English clubs, meanwhile, will have home advantage in the Amlin Challenge Cup quarter-finals.
Top seeds Bath entertain Stade Francais, with Gloucester hosting Biarritz and Wasps meeting Leinster. The fourth tie sees French heavyweights Perpignan and Toulouse going head to head.
Wasps rugby director David Young said: "You could look at it and say it's not much reward for topping our pool to then be given a draw against the Heineken Cup champions.
"But on the flip side, what a hugely exciting prospect it will be playing them at home. It is a game which, if we can win it, would really put us on the map."
- Sports & Recreation