Anwar's acting debut was in the British miniseries Hideaway. She made her film debut in Manifesto, which was followed by more British television productions including First Born, Summer's Lease, Press Gang, and The Mysteries of the Dark Jungle. While working on movies and television in London, she met American actor Craig Sheffer, and moved with him to Hollywood. Anwar and Sheffer have co-starred in a number of American movies together, such as In Pursuit of Honor, The Grave, Turbulence 3: Heavy Metal, Flying Virus, Save It for Later, Water Under the Bridge, and Long Lost Son.

Her first American film was If Looks Could Kill. She followed that with the acclaimed films Wild Hearts Can't Be Broken, Scent of a Woman, Body Snatchers, For Love or Money, and The Three Musketeers. In 1993, People magazine named her one of the 50 most beautiful people in the world.

One of her most memorable moments on screen came in 1992's Scent of a Woman when she danced a tango with Al Pacino,