Kelly made her Broadway debut in 1928. In 1930 and 1931, she performed for Producer Earl Carroll in his popular Sketches and Vanities musicals.

At a time when being openly gay was not socially acceptable, Kelly was open about her sexuality. On occasion she would frankly disclose, in public and with typical candor, to being a "dyke". During the 1930s, she disclosed to Motion Picture magazine that she had been living with Actress Wilma Cox for several years and had no intention of getting married. She later claimed she had an affair with Tallulah Bankhead when she worked as Bankhead's personal assistant.

In August 1933 Kelly was injured as a Passenger in a sedan driven by Gene Malin, the prominent drag performer. Malin apparently confused the gears and reversed off a pier into the water, after leaving a club in Venice, Los Angeles. Kelly suffered serious injuries.

After the popularity of shorts began to wane, Kelly moved to full-length feature films, often playing working-class character roles in comedies and musicals. One of her memorable roles was as Etta, the cook, in the five Academy Awards-nominated 1938 comedy movie Merrily We Live.

After her film career declined in the mid-1940s, Kelly returned to New York where she worked in radio and summer stock. She also became a lifelong friend, personal assistant, and frequent lover of Tallulah Bankhead. Kelly returned to the screen after 17 years with guest spots on television and in film roles.

Kelly was born Sarah Veronica Rose Kelly in Brooklyn to Irish immigrants parents John (died 1942) and Delia Kelly (1875-1930). She began her career in vaudeville as a Dancer at the age of 12. While in vaudeville, she performed in Frank Fay's act, first in a song-and-dance routine and later as Fay's comic foil. In one routine, Kelly told Fay and the audience that she had been at the beauty parlor. Fay remarked, "And they didn't wait on you?" She remained with Fay for several seasons until Fay eventually dismissed her.

By 1943, Kelly's film career had begun to decline. She appeared in films for Producers Releasing Corporation, the smallest and cheapest of the movie studios. Her last starring roles were in two PRC comedies, My Son, the Hero and Danger! Women at Work, both released in 1943. Kelly left Hollywood and would not make another film for 17 years.

Kelly returned to the screen in the 1950s with television and sporadic film roles. On television she appeared in guest roles on 26 Men, Kraft Television Theatre, The Man from U.N.C.L.E, The Dick Van Dyke Show, The Wild Wild West, and Alfred Hitchcock Presents, as well as many unsold pilots. She also made a memorable appearance as Laura-Louise in the film thriller Rosemary's Baby (1968), directed by Roman Polanski, alongside veteran actors Sidney Blackmer, Ruth Gordon, and Maurice Evans.

She returned to Broadway in 1971 in the smash revival of No, No, Nanette with fellow hoofers Ruby Keeler and Helen Gallagher. Kelly scored a huge success as the wisecracking, tap-dancing maid, and won Broadway's 1971 Tony Award for Best Featured Actress in a Musical for her performance in the show. She matched that success the following year when she starred in Irene with Debbie Reynolds, and was again nominated for a Tony.

In 1976, she appeared as the housekeeper Mrs. Schmauss in the film Freaky Friday starring Jodie Foster and Barbara Harris. Her final movie appearance was in the 1979 Disney comedy The North Avenue Irregulars, also co-starring Harris, along with Cloris Leachman, Edward Herrmann and Karen Valentine. Kelly's last onscreen appearance was a guest spot in a two-part episode of The Love Boat in 1979.

In January 1980, Kelly suffered a stroke while in San Francisco which caused her to lose the ability to speak. She was admitted to Englewood Nursing Home in Englewood, New Jersey, on the advice of her old friend Ruby Keeler where she underwent therapy.

On September 24, 1981, Kelly died of cancer at the Motion Picture & Television Country House and Hospital in Woodland Hills, California. She is interred alongside her parents, John and Delia Kelly, in Calvary Cemetery in Queens.