The statement, posted on Caan's official Twitter account, said the actor died on Wednesday evening (local time). No cause of death or location were provided.
"The family appreciates the outpouring of love and heartfelt condolences and asks that you continue to respect their privacy during this difficult time," the statement said.
Caan was nominated for an Academy Award for best supporting actor for his portrayal of the hot-tempered Corleone in 1972's "The Godfather." He reprised the role in flashback scenes in "The Godfather: Part II" in 1974. Other nominations included four Golden Globes and an Emmy.
Al Pacino, who played Sonny Corleone's brother Michael in "The Godfather," praised Caan as "a great actor, a brilliant director and my dear friend."
"It’s hard to believe that he won’t be in the world anymore because he was so alive and daring," Pacino said in a statement.
Francis Ford Coppola, who directed Caan in "The Godfather" and 1969 movie "The Rain People," said Caan's films "and the many great roles he played will never be forgotten."
Caan's career spanned six decades and included a broad range of other roles in movies from psychological thriller "Misery" to comedy "Elf." Last year, he starred in romantic comedy "Queen Bees" opposite Ellen Burstyn.
James Edmund Caan was born in the Bronx borough of New York City on March 26, 1940 to German Jewish immigrants.
One of three siblings, Caan began acting on television in 1961.
He gained widespread acclaim for his role in the 1971 TV movie "Brian's Song," in which he played real-life Chicago Bears football player Brian Piccolo, who died of cancer at age 26. The role earned Caan an Emmy nomination.
Billy Dee Williams, who starred with Caan in "Brian's Song," shared an undated photo of the two together in recent years on his Twitter account.
"Team Mates and friends till the end. RIP Jimmy," Williams tweeted with a broken heart emoji.
