Davenport, the son of actors Nigel Davenport and Maria Aitken, was born in Suffolk, England and lived in Ibiza for the first seven years of his life. His uncle is writer and politician Jonathan Aitken, his maternal grandfather was politician William Aitken and his maternal great-grandfather was the first Baron Rugby. His parents divorced when he was seven, at which point he was sent to The Dragon School, a boarding school in Oxford, as his parents did not want him to become involved in the divorce proceedings. He then went on to attend Cheltenham College and then the British American Drama Academy. He is married to the actress Michelle Gomez.

Davenport had not planned to become an actor, however, his career began when he took a gap year after attending Cheltenham College. A director from Clwyd Theatr Cymru (The Welsh National Theatre) was impressed by his performance on a summer drama course and asked Davenport to work for him. At 18 years of age he ended up in Wales, performing bit parts in Hamlet, where he became friends with Rhys Ifans(Notting Hill).

The following year, he attended the University of East Anglia (UEA), concentrating on Film Studies and English Literature. He tried out acting once more at the UEA, but was not, at this point, seriously interested in it. In fact, Davenport had considered becoming a member of a film crew, as opposed to acting in front of the camera. His mother advised him, after his graduation from UEA, to write to John Cleese requesting work on the set of his upcoming film, Fierce Creatures, so that Davenport could gain some real experience behind the camera. Cleese instead sent Davenport's letter to the casting department, and he was cast as a student zookeeper. A small part with very few lines provided his first opportunity to work in front of the camera. After the production of Fierce Creatures was completed, Davenport found an agent, who secured an audition for the role of Miles Stewart in the BBC television drama series This Life. It was in this part that Davenport first rose to fame.

Since then Davenport has played roles in many successful films and TV series, including The Talented Mr. Ripley, Coupling, and Ultraviolet, as well as the box office smash Pirates of the Caribbean: The Curse of the Black Pearl, and its sequels, Pirates of the Caribbean: Dead Man's Chest and Pirates of the Caribbean: At World's End.

During his career Davenport has also performed voice-overs, having narrated the audio versions of John Buchan's The Thirty-Nine Steps and recorded parts in Anthony Burgess' A Clockwork Orange. He also provides the voice over for the British MasterCard advertisements.

In 2006, he featured in the ITV1 drama The Incredible Journey of Mary Bryant and in 2009 he will star in the film The Boat That Rocked.

On 2 December 2008, TV Guide reported that Davenport has been cast as Lloyd Simcoe, a father who labors to get to his hospitalized son in the new Abcpilot Flash Forward, which is based on a Robert J. Sawyer novel and is said to be a possible “companion show” to Lost.