She was born in New York, the daughter of Dancers Isabel Mirrow and Kelly Kingman Brown (1928-1981). Her mother Isabel died in 2014. Her middle name, Sue, was the name of her maternal grandmother, Sue Brown, a respected dance Teacher in Mississippi. She has two brothers and one sister; her brother Kevin is a film Producer. At the age of seven she began dancing, and would be trained at her father's studio in Arizona, along with her brother Ethan and her sister Elizabeth. She earned a scholarship to study at the School of American Ballet, then joined the distinguished New York City Ballet. She added an "e" to her last name for her stage name considering it more feminine after being mistaken as male in a Playbill.

In 1976 she joined the American Ballet Theatre as a soloist, then became principal in 1986. She retired from the company in 1993. Since then she has made guest appearances, studied acting for three years, and made her Broadway debut in the show The Red Shoes. She has also been choreographing and teaching dancing. In 1997 she was awarded the Distinguished Achievement Award by the New York City Dance Alliance.

In 1977, a film based on her family, The Turning Point was created by her godmother, ballerina Nora Kaye and her husband, the Director Herbert Ross. Originally, Gelsey Kirkland was to play Leslie, but after Kirkland dropped out due to substance abuse issues as well as a dislike of the script, Ross cast Browne in the role, believing she could play a fictionalized version of herself. She went on to be nominated for the Golden Globe and Academy Award for Best Supporting Actress.

Browne also appeared in the dance films Nijinsky (1980) and Dancers (1987), each directed by Herbert Ross. She also appeared on the television series Happy Days as a special guest star as a dancer-girlfriend of Fonzie's.