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, Burrows was discovered by a modelling talent scout (she stands about 6 feet tall), and she began a successful modelling 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.

In Jan 2005 she created the role of Janey in the world premiere of Earthly Paradise at the Almeida Theatre and 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.