Cochrane was born the third child (siblings Brendan and Siobhan) to a Scots-Irish American father and an Indian American mother. He spent his childhood in Grantchester, England. Cochrane then returned to the United States and was educated in New York City and trained in the drama department of NYC's Fiorello H. LaGuardia High School of Music & Art and Performing Arts. His first roles included a part in a docudrama about drugs on Saturday Night with Connie Chung in 1989 and an appearance in an episode of H.E.L.P. in 1990. He then made his film debut (with about fifteen seconds screen time) in A Kiss Before Dying, followed by his first substantial role as Jeff Goldblum's son in Fathers and Sons. Cochrane played major roles in the films Empire Records and Dazed and Confused, and appeared in Hart's War starring Bruce Willis. Most recently he appeared in the Richard Linklater film A Scanner Darkly (2006).

He also appeared in the TV drama CSI: Miami from 2002-2004.

Although his private life is closely guarded, it is known that he had about a two year relationship with Renée Zellweger and has dated Liv Tyler. He is friends with Ben Affleck and Joaquin Phoenix and is close friends also with his former CSI: Miami co-star Adam Rodriguez.

He enjoys travelling, likes to play chess, and is a passionate motorcycle rider. In photos he almost always appears unshaven.