Anwar was born in Laleham, Middlesex, England. Her father is Tariq Anwar, an Iranian born award-winning British film editor of Iranian and Austrian parents, and her mother is an English actress. Anwar attended Laleham C. of E. Primary and Middle School from 1975 to 1982; an end-of-term St. Trinians sketch in the school concert of 1982 gave an early indication of her theatrical leanings. She studied at a London drama and dance school.
Anwar has a daughter, Willow, born in 1993, with whom she appeared in the film Daddy Who?. Willow's father is American actor Craig Sheffer with whom Anwar had a lengthy relationship in the late 1980s and 1990s. Anwar was later married to actor John Verea; the couple had a son and daughter together before divorcing. Additionally, Anwar had a relationship with actor Johnathon Schaech in 1997.
In June 2007 Anwar announced during an interview regarding Burn Notice that she was a pagan.