West was born in Sheffield, Yorkshire, England to George West, who owned a plastics manufacturing plant, and Moya, a housewife. He has described himself as being from an Irish Catholic middle class family. Dominic attended Eton College and Trinity College, Dublin for his education.
West made his debut in 1991 in a short film called 3 Joes. His other screen appearances include True Blue, Chicago and Mona Lisa Smile.
In film, he portrayed the heavy-metal guitarist Kirk Cuddy in the 2001 film Rock Star. In 2007, he played the Spartan politician Theron in 300.
As a theatre actor, West has appeared as Konstantin in the Peter Hall-directed The Seagull (1996) at the Old Vic. In 2005, he returned to the stage to play Orlando in David Lan's As You Like It at the Wyndhams's Theatre in London. He played Edward in Harley Granville Barker's The Voysey Inheritance directed by Peter Gill at the Royal National Theatre in 2006. In late 2006 and early 2007 he played the lead role of Jan in Tom Stoppard's Rock 'n' Roll in London's West End.