Burrows made her film debut in 1993 with a small role in Jim Sheridan's "In the Name of the Father". Her first significant acting roles came in 1995, as an ambitious young Irishwoman in "Circle of Friends". She appeared in Ngozi Onwurah's "Welcome II The Terrordome". Following this, she starred in the BBC production of Dennis Potter's "Karaoke" (1996). Subsequently, she appeared in "Hotel de Love", "Lovelife", "Nevada", "One Night Stand", and "The Matchmaker". In 1999 she appeared in Mike Figgis' experimental film "The Loss of Sexual Innocence", in which she played twins: one raised in England, the other in Italy. Burrows appeared in the thriller film "Deep Blue Sea" and later played the title role in a film of "Miss Julie", which premiered at the Toronto International Film Festival.