The Premiership side fought hard in a cagey opening period against their Top 14 adversaries as the sides went in at 6-6 after 40 minutes, Morne Steyn of Stade and Nick Evans of Harlequins both landing two penalties each.
But Quins stormed into a lead early in the second half thanks to a converted try from Mike Brown.
Evans also kept the scoreboard ticking over with two second-half penalties and a drop goal, while Tim Molenaar scored a second try three minutes from time to put the game beyond any doubt.
Harlequins will play Northampton in their semi-final on April 25 at Franklin's Gardens.
- Sports & Recreation
- Nick Evans
- Morne Steyn