The 34-year-old was among the starters as Claudio Ranieri's charges slipped to a 3-1 semi-final defeat after extra time against Ligue 1 strugglers Guingamp at the Stade du Roudourou.
Monaco - who sit second in Ligue 1 - had started the game as strong favourites against 17th-placed Guingamp, but failed to live up to expectations in a lacklustre showing.
Abidal failed to hide his annoyance after the game, especially considering Monaco had targeted glory in the Coupe de France as a key objective.
"The team is very disappointed," he told Monaco's official website. "We have been really keen to win this game.
"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 will look to get back on track in Sunday's Ligue 1 encounter with Nice.
'I can actually assure you Mohamed Salah has been offered a contract! I'd also be looking at them like, why haven’t you signed? The team is made for them, where are they gonna go?': Liverpool icon shares view on star's situation
Arsenal report: Incredible Roberto Firmino deal 'close', following recent Mikel Arteta 'hint'
