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.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.