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Stade Francais travelled to Harlequins memorial ground, and thrashed the home side by 26 points to nil.
Tries came from full back Arthur Gomes and outside half Brian Liebenberg.
The ever consistent boot of Diego Dominguez added two conversions, and three penalty goals, with Liebenberg added a third penalty to compound Harlequins misery.
Harlequins now have a lot of work to do in next week's return leg if they are to make it through to the quarter-finals.
Borders return to Scotland with a 19-16 Parker Pen Challenge Cup loss to Montauban in the record books, but will be confident of over overturning the three-point deficit on home soil.
Borders scored tries through backs Scott Paterson and Jono Stuart, but let an 11-0 lead slip. Two dropped goals to outside-half Sebastien Fauque proved crucial in the final quarter.
In Narbonne, Irish side Connacht crashed 27-42 but showed enough to return to their tiny Galway Sportsground with some confidence next weekend.
Narbonne fullback Pascal Bazani picked up a hat-trick of tries, while outside-half Jerome Gendre added 14 points.
For Connacht, winger Wayne Munn bagged a brace of tries and reliable goal-kicker Eric Elwood added 12 points to give his side a chance in the second leg next weekend.
