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Bordeaux will host Racing 92, which earlier ran in seven tries against Edinburgh in a 56-3 romp, in one of two all-French quarter-finals next weekend. In the other, Clermont will host Toulouse.
A fifth French team, La Rochelle, will be at home to Sale.
Defending champion Exeter, which came from 14 points behind to beat Lyon, will meet Leinster, which was given a walkover against Toulon after a Covid-19 positive in the French club's squad.
In Bordeaux, the English Premiership leader started strongly when winger Henry Purdy touched down after seven minutes.
Fly-half Callum Sheedy missed the conversion. Although he kicked three penalties before half-time his counterpart, France international Matthieu Jalibert, had put the home team ahead by half-time with five penalties.
Jalibert then burrowed over the line from close range to add a try six minutes into the second half.
Purdy came close to a quick reply, but Romain Buros stripped the ball as the Bristol winger went to touch down.
Sheedy closed the gap with a penalty but Bordeaux-Begles locked up the victory in the last 10 minutes, even though lock Cyril Cazeaux was shown a yellow card in the 77th minute.
South African hooker Joseph Dweba touched down with 10 minutes left and Nans Ducuing ran back an interception for a final try with a minute left.
In Paris, after a couple of early scares, Racing, coming off two straight losses in the Top 14 and without suspended Scotland star Finn Russell, took control.
"Really proud effort from our boys today," Racing fullback Kurtley Beale told British broadcaster BT Sport. "Great way to turn round from a disappointing loss in the Top 14 last week."
Hooker Camille Chat crashed over under the posts for the first Racing try after 28 minutes.
Five minutes before the break, Racing carved Edinburgh open with pace, power and slick handling.
Centre Virimi Vakatawa drew a defender and flipped a backhanded pass to the surging No.8 Jordan Joseph on the wing.
He passed inside to Louis Dupichot and the centre immediately flipped the ball on to scrum-half Maxime Machenaud who trotted over for a try.
Joseph bludgeoned over after 64 minutes to start a burst of five tries in the final 16 minutes.
Prop Guram Gogichashvili, France winger Teddy Thomas, twice, and veteran back Francois Trinh-Duc all touched down in the late scoring burst.
"We fired some early shots," Edinburgh coach Richard Cockerill said. "But we lost out on the physical battles. Ultimately, we got what we deserved, they're too good a side."
Sale reached the quarter-finals for the first time in 15 years after crushing the Scarlets 57-14.
Hooker Akker van der Merwe scored two first-half tries, while fly-half AJ MacGinty kicked two conversions and three penalties for a 23-0 lead at half-time.
MacGinty finished with 32 points, with his haul including a try.
Winger Marland Yarde, lock Josh Beaumont and substitute scrum-half Raphael Quirke also touched down.