Her best-known films include 1938's Quai des Brumes (Port of Shadows), in which she starred alongside Jean Gabin.
She also appeared in Carol Reed's The Fallen Idol and opposite Humphrey Bogart in Passage to Marseille.
In a statement, French president Francois Hollande said she "personified elegance and grace".
He said: "Michele Morgan was more than just a gaze. Her legend left its mark on many generations."
Morgan's Cannes award came in 1946 for her role as a blind woman in La Symphonie Pastorale (Pastoral Symphony).
In later life, she received lifetime achievement accolades at the Venice Film Festival and the Cesar Awards, as well as a star on the Hollywood Walk of Fame.
Morgan was considered for the role of Ilsa Lund in Casablanca, but lost out to Ingrid Bergman when film studio RKO refused to loan her to Warner Bros.
Her film career faded with the arrival of the French New Wave in the 1960s, although she continued to work in television and on stage.
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Michele Morgan, the French screen star and glamour icon who won the first best actress prize at the Cannes film festival, has died at the age of 96.