Brennan was born as Verla Eileen Regina Brennan to an Irish American Catholic family; she is the daughter of silent actress Jean Manahan.

Although it is not certain as to when she began her acting career, it is known that her beautiful soprano voice enabled her to play the title character in the off-Broadway musical/operetta Little Mary Sunshine (1959) as well as the role of Irene Malloy in the original Broadway performance of Hello, Dolly! (1964). Her feature film debut was in Divorce American Style (1967).

She soon became one of the most recognizable (if not unnamable) supporting actresses. Her roles were usually sympathetic characters, though she has managed to play a variety of other character types, including earthy, vulgar, and sassy. One year after her feature film debut, she became a semi-regular on the comedy-variety show Rowan & Martin's Laugh-In, where she stayed for only two months.

Although she had not been able to have her name recognizable to the general public, she had become a favorite of many directors, including Peter Bogdanovich. Bogdanovich was the only director who made use of her musical talents (before, she sang in performances off Broadway) when he cast her as Cybill Shepherd's crude, fun-loving maid in his musical flop At Long Last Love (1975).

She also proved to be somewhat of a favorite of director Robert Moore and writer Neil Simon, when she starred in Murder by Death (1976) and The Cheap Detective (1978).

In 1980, she received an Oscar nomination for her role as Goldie Hawn's nasty commanding officer in Private Benjamin; she reprised the role in the television adaptation (1981-1983), for which she won an Emmy. While she has been cast in a number of similar "cranky" parts, her role in White Palace (1990) (as Susan Sarandon's older sister) showed her more sympathetic side.

In 1984, she was seriously injured in a car accident; she took three years off work to recover. During this time, her performance as Mrs. Peacock in Clue (1985) reached theatres.

In recent years, she has guest starred in television series, including recurring roles as the nosey Mrs. Bink in 7th Heaven, and as the gruff acting coach Zandra in Will & Grace.

From 1968 to 1974 she was married to David John Lampson, with whom she has two sons: Patrick (a basketball player) and Sam (a singer).

She is a breast cancer survivor.