Gavron was born in Hendon, Middlesex, the son of American-born writer Martha Pichey and the late publisher Simon Gavron. His paternal grandparents were Robert Gavron, Baron Gavron and Hannah Fyvel. He has two younger brothers; Benjamin (born 1992) and Moses (born 1996).

As Miro, a young burglar who uses techniques from parkour to gain access to the high-tech offices of an urban landscape architect, he performs several difficult physical feats in the film. For his role, Gavron received a nomination for Most Promising Newcomer from the British Independent Film Awards (BIFA).

Gavron played the hired assassin named Duro in the second season of HBO and BBC production, Rome. He also appeared in the 2008 release Nick and Norah's Infinite Playlist as Dev, the gay lead singer of Nick's band.

In Inkheart, a movie adaptation of a novel originally titled Tintenherz, Rafi is set to play the role of a boy named Farid . Farid, a character from The Book of One Thousand and One Nights is magically brought to life and crushes on Meggie, the daughter of a bookbinder named Mo, who has the ability to bring characters from books into the real world when he reads aloud. The film was due to be released theatrically in spring of 2008, but has been pushed back to January 2009.