Brennan's managers, Jessica Moresco and Al Onorato, said she died Sunday at home in Burbank after a battle with bladder cancer.
"Our family is so grateful for the outpouring of love and respect for Eileen," her family said in a statement. "She was funny and caring and truly one of a kind. Her strength and love will never be forgotten."
Brennan got her first big role on the New York stage in "Little Mary Sunshine," a musical comedy that won her the 1960 Obie award for best actress. Along with her "excellent singing voice," her performance was "radiant and comic," said a New York Times review.
But it was a series of sharp-tongued roles that won her fans on television and in movies, including gruff Army Capt. Doreen Lewis in 1980's "Private Benjamin," aloof Mrs. Peacock in 1985's "Clue" and mean orphanage superintendent Miss Bannister in 1988's "The New Adventures of Pippi Longstocking."
"I love meanies, and this goes back to Capt. Lewis in 'Private Benjamin,'" Brennan said a 1988 interview with The Associated Press. "You know why? Because they have no sense of humor. People who are mean or unkind or rigid - think about it - cannot laugh at themselves. If we can't laugh at ourselves and the human condition, we're going to be mean."
"Private Benjamin" brought her a supporting actress nomination for an Oscar. She also won an Emmy for repeating her "Private Benjamin" role in the television version and was nominated six other times for guest roles on such shows as "Newhart," ''thirtysomething," ''Taxi" and "Will & Grace."
"Our world has lost a rare human," said "Private Benjamin" star Goldie Hawn in a statement. "Eileen was a brilliant comedian, a powerful dramatic actress and had the voice of an angel. I will miss my old friend."
Brennan's "Private Benjamin" role led to an enduring friendship with Hawn. A couple of years after they filmed the movie, Brennan and Hawn had dinner one night in 1982 in Venice, Calif. As they left the restaurant, Brennan was struck by a car. Her legs were smashed, bones on the left side of her face were broken, her left eye socket was shattered. Brennan said she fought her injuries with rage.
"I was no saint," she said in an interview with Ladies Home Journal. "I was angry, and anger is a powerful emotion. It increased my determination not to go under, to get well."
Brennan became dependent on painkillers, and two years after the accident she entered the Betty Ford Center to cure her addiction.
"We get addicted to dull the pain of life," she told the magazine. "But once we accept that life is tough and painful, we can move on and grow and evolve."
A decade after the accident, she said she was glad she was struck by the car.
"You learn from powerful things," she said in 1992. "Initially, there's enormous anger, but your priorities get shifted around."
Brennan was a member of the original company of "Hello, Dolly" on Broadway. From the New York stage, she moved to the screen in "Divorce American Style" and "The Last Picture Show," a pair of appearances on "Rowan & Martin's Laugh-In," and TV guest shots on everything from "All in the Family" and "McMillan & Wife" to "Kojak," ''The Love Boat," ''Murder She Wrote" and "Mad About You."
Brennan was born Verla Eileen Regina Brennan in Los Angeles. She was educated in convent schools and studied at Georgetown University and the American Academy of Dramatic Arts in New York.
She is survived by her two sons, Sam and Patrick Brennan.