Fletcher died in her sleep surrounded by family at her home in Montdurausse, France, her agent David Shaul told The Associated Press on Friday.
No cause of death was given.
After putting her career on hold for years to raise her children, Fletcher was in her early 40s and little known when chosen for the role opposite Jack Nicholson in the 1975 film by director Milos Forman. At the time, she didn't know that many other prominent stars, including Anne Bancroft, Ellen Burstyn and Angela Lansbury, had turned it down.
"I was the last person cast," she recalled in a 2004 interview.
"It wasn't until we were halfway through shooting that I realised the part had been offered to other actresses who didn't want to appear so horrible on the screen."
The character was so memorable she would become the basis for a Netflix series, Ratched, 45 years later.
Cuckoo's Nest went on to become the first film since 1934's It Happened One Night to win Best Picture, Best Director, Best Actor, Best Actress and Best Screenplay.
Clutching her Oscar at the 1976 ceremony, Fletcher told the audience, "It looks as though you all hated me."
She then addressed her deaf parents in Birmingham, Alabama, talking and using sign language: "I want to thank you for teaching me to have a dream. You are seeing my dream come true."
She was nominated for Emmys for her guest roles on the TV series Joan Of Arcadia and Picket Fences, and had a recurring role as Bajoran religious leader Kai Winn Adami in Star Trek: Deep Space Nine. She played the mother of musical duo Carpenters in 1989's The Karen Carpenter Story.
Fletcher married producer Jerry Bick in the early 1960s and gave birth to two sons in quick succession. She decided to put her career on hold to be a stay-at-home mother and didn't work for 11 years.
"I made the choice to stop working, but I didn't see it as a choice," she said in the 2004 interview. "I felt compelled to stay at home."
She divorced Bick in 1977 and he died in 2004.
Watching movies as a child convinced Fletcher she wanted to act. She was further inspired, she once said, when she saw the movie Lady In The Dark with Ginger Rogers.
That and other films, Fletcher said, taught her "Your dream could become real life if you wanted it bad enough."
"I knew from the movies," she would say, "that I wouldn't have to stay in Birmingham and be like everyone else."
Fletcher's death was first reported by Deadline.
She is survived by her two sons, John and Andrew Bick.
