The fourth-tier side even survived the late dismissal of Ali Keita to claim an upset and move into the last 32 of the country's biggest cup competition.
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The Championnat de France outfit are three divisions below their opponents, but managed to withstand heavy pressure before nicking a goal on the hour.
Without absent forward Jordan Ayew, the visitors to the 800-capacity Stade Rene Fenouillere did not have the firepower to find a reply against opposition whom they were expected to overturn with ease.
Lorient could not even find a reply after Keita saw red for an 83rd-minute lunge on Pedrinho, as Avranches held on for a famous win.