Though best known for her supporting performances in films, Dillon got her start as an improvisational comedian and stage actress. Her first major role was as Honey in a 1962 Broadway production of Edward Albee's Who's Afraid of Virginia Woolf, for which she was nominated for a Best Supporting Actress (Dramatic) Tony Award.

She followed her early Broadway success with her first film, The April Fools, in 1969. Playing "Memphis Sue" opposite David Carradine, she was nominated for the Best Female Acting Debut Golden Globe for the 1976 Woodie Guthrie biopic Bound for Glory. She was nominated for a Best Supporting Actress Oscar for the role of a mother whose young child is abducted by aliens in Steven Spielberg's Close Encounters of the Third Kind in 1977. Four years later she was once again nominated for a Best Supporting Actress Oscar for her performance as a suicidal teacher in 1981's Absence of Malice, opposite Paul Newman and Sally Field.

As a comedian, she is perhaps best known for her role as the compassionate mother of Ralphie in Bob Clark's 1983 film A Christmas Story. The film was based on a series of short stories and novels written by Jean Shepherd, and follows young Ralphie Parker (played by Peter Billingsley) on his quest for a rifle from Santa Claus. Five years later she appeared opposite John Lithgow in the Bigfoot comedy Harry and the Hendersons. She continued to be active in stage and film throughout the 1990s, taking minor roles in the Barbra Streisand drama The Prince of Tides, the low-budget Lou Diamond Phillips thriller Sioux City, and the drama How to Make an American Quilt.

She has remained a private person, and information about her personal life is largely unknown. She was married briefly to character actor Richard Libertini, with whom she had one child. In recent years her career has waned, but she has taken notable roles in the 1999 ensemble drama Magnolia and the TV adaptation of John Grisham's A Painted House in 2003.