The Ligue 1 strugglers will have been hoping to make a flying start to the new year in Normandy, however, they instead came unstuck against the Championnat National outfit, who prevailed thanks to a 60th-minute strike from Maguette Diongue.
Avranches held on in the closing stages despite seeing defender Ali Keita sent off nine minutes from time.
Lorient were not the only top-flight side to suffer an early exit, with Nice also tumbling out courtesy of a 2-0 defeat to Ligue 2 outfit Valenciennes, which came courtesy of first-half strikes from Adama Coulibaly and Rydell Poepon.
There were no such problems for three-time winners Nantes, as Michel Der Zakarian's men cruised to a 4-0 win over 10-man Club Franciscain, while Reims and Evian also progressed.
The all-Ligue 1 tie between Bastia and Lille at Stade Armand Cesari was settled in the hosts' favour thanks to Floyd Ayite's double, while Reims claimed a 3-1 success at St Maur Lusi, and 2013 finalists Evian eased to a 3-0 triumph at Bobigny.
Five-time winners Red Star beat Arles 2-1, but the most entertaining tie came at the Stade de la Liberation as midfielder Xavier Mercier scored four goals in Boulogne's 5-4 victory versus Sarre-Union.
Iris Club de Croix were the biggest winners of the day, hammering Saint-Omer 5-0 in northern France.
Ligue 2 sides Tours, Brest and Chateauroux all progressed, but Niort were 1-0 losers at fourth-tier Concarneau.
Elsewhere, there were wins for Jura Sud, Andrezieux, Bressuire, Le Poire-sur-vie and Quevilly, who beat Orleans 5-3 on penalties after extra time ended in a 3-3 draw.
