Epinal, who eliminated Lyon in the previous round, outplayed Nantes who lie second in the Ligue 2 standings.
Judicael Crillon opened the scoring for the hosts nine minutes into the second half, only for Filip Djordjevic to equalise in the last minute and send the game into extra time.
No goals were scored and goalkeeper Olivier Robin stopped two penalties in the shootout as Epinal won 4-3 to go through.
"It's crazy. Everyone gave everything, we're all extremely tired but we won," midfielder Valentin Focki told Eurosport.
Minnows Mende's French Cup run ended in abrupt fashion when the amateurs were humbled by second division side Le Havre.
Mende, who play in the Division d'Honneur (seventh division) and eliminated second division Arles Avignon in the previous round, held Le Havre until 10 minutes from time.
Alexandre Bonnet broke the deadlock before Walid Mesloub and Elhadji Ba wrapped it up.
Ligue 1 side Evian Thonon Gaillard struggled but went through with a 2-0 win at fifth division side Vertou and Lorient beat Ligue 2 Sedan 1-0.
Troyes had less trouble, seeing off fourth division side Fontenay Le Comte 5-0.
The Rouen v Marseille game was postponed because of a non-toxic gas leak from a chemical factory.
Paris St Germain take on Toulouse in an all Ligue 1 clash on Wednesday.