Heavy rain in Correze rendered the Stade Amedee-Domenech surface unplayable, but Stade Alexandre-Cueille in nearby Tulle stepped in to host the match later in the day.
The bonus-point victory saw Brive dislodge Worcester to become one of three best-placed runner-up, finishing nine points behind Pool 5 winner Connacht, which thumped Oyonnax 50-14.
It was a similar story for Stade Francais, whose 17-10 win against Edinburgh earned it a place alongside the Pool 4 leader in the last eight, while London Irish finished third despite a 24-17 win against Krasny Yar.
A 33-17 win for the Dragons over Bordeaux b was not enough to see them through, with Stade snatching its spot in the final match of the day, and there was a 33-19 win for Pool 1 victor Newcastle Falcons against Enisei STM.
Cardiff Blues were already assured of winning Pool 2 and finished off with a 21-18 victory at Lyon, but they will be the group's only representatives in the last eight after Sale Sharks lost 28-21 in Toulouse.
The day's remaining clash saw Zebre beat Agen 38-30 in Pool 3, with Pau and Gloucester having already taken the top two spots.
European Challenge Cup quarter-final draw:
Pau v Stade Francais
Edinburgh v Cardiff Blues
Connacht v Gloucester
Newcastle Falcons v Brive