Brennan's managers, Jessica Moresco and Al Onorato, said she passed away 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 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 7TH HEAVEN, ER, ALL IN THE FAMILY, THIRTYSOMETHING, WILL & GRACE 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.