Monaco - who sit second in Ligue 1 - went into the last four tie as strong favourites against their 17th-placed opponents, but failed to live up to expectations in a lacklustre showing at Stade de Roudorou.
"The team is very disappointed," Abidal told Monaco's official website. "We have been really keen to win this game.
The defeat looks set to condemn Monaco to a trophyless season with Claudio Ranieri's side also trailing league rivals Paris Saint-Germain by 10 points in Ligue 1 with five games to go.
The 34-year-old added: "The Coupe de France was a goal we had since the beginning of the season. We were on the verge of getting there.
"It is very hard to lose. We have two goals, one ended tonight."
Monaco return to domestic action with Sunday's Ligue 1 fixture at home to Nice.