Epinal had knocked out top flight Lyon and Nantes in earlier rounds, but their giant-killing run came to an end in heartbreaking fashion as they gave away a ninth-minute penalty - scored by Pierrick Valdivia - and fell 2-0 behind after an own goal by defender Calvin Mangan in the 21st minute.
Lens missed a penalty in the second half but held on to complete a win.
They go on to play Bordeaux for a place in the semi-finals.
Coupe de France quarter-final draw:
(Ligue 1 unless stated)
RC Lens (II) v Bordeaux
Troyes v Nancy
St Etienne v Lorient
Evian Thonon Gaillard v Paris St Germain