Burrows was born in London, England, to a politically active family; both of her parents are socialists. Her father is an architect and teacher and her mother, Susie, is a teacher and feminist. At age fifteen, she was discovered by a modeling talent scout (she stands about 6 feet tall), and she began a successful modeling career.

Burrows debuted in the film In the Name of the Father (1993). Her first significant acting role was as an ambitious Irish lass in Circle of Friends (1995), which also featured Chris O'Donnell and Minnie Driver. Subsequently she appeared in Hotel de Love. In 1999, she appeared in the sci-fi Wing Commander (1999), the thriller Deep Blue Sea, and The Loss of Sexual Innocence. The following year she appeared in Miss Julie and Timecode.

In 2001, she had roles in Enigma, and Tempted with Burt Reynolds, Peter Facinelli and Michael Arata. In 2004 she played the part of Andromache in the big-budget movie Troy. On 30 October 2005 she appeared on stage at the Old Vic theatre in London in a 24-hour play, Night Sky, alongside Christopher Eccleston, Bruno Langley, David Warner, Navin Chowdhry and David Baddiel.

As of 2007 Burrows plays attorney Lorraine Weller on ABC's Boston Legal (Season 4).

In 2008, she appeared in the independent film, The Guitar, which made its debut at the Sundance Film Festival, as well as in a starring role in the heist film The Bank Job.

Burrows is a close friend of politician Tony Benn. She is also a socialist and has spoken of her admiration for French Socialist politician Ségolène Royal. She joined an anti-racism group when she was 11 years old and she went on to become the Vice President of the National Civil Rights Movement.

In a 1999 interview with Film Unlimited, Burrows revealed that she is bisexual. For a number of years, she was involved with film director Mike Figgis, and she starred in some of his films, including Miss Julie - one of the first films to experiment with multi-camera views, and Figgis' quad-screen film, Timecode.