Brennan's managers, Jessica Moresco and Al Onorato, said she died at home 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.
She went on to win fans for her sharp-tongued roles 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 AP. "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 Best Supporting Actress nomination for an Oscar. She won an Emmy for repeating her Private Benjamin role in the TV version, and was nominated six other times for guest roles on such shows as Newhart, thirtysomething, Taxi and Will & Grace.
"We have lost my old friend Eileen Brennan. No one ever made me laugh more!" Private Benjamin star Goldie Hawn Tweeted. "Now I cry. Please keep singing darling from on high. Rest."
Brennan's Private Benjamin role also 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.
Shifting Priorities"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 in 1984. "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 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.