Green is the son of saxophonist Benny Green and the actress Toni Kanal. Dominic reads English Literature at St John's College, Oxford. After graduation, he worked as a professional jazz guitarist, playing with his father, John Dankworth, Doris Troy, Kym Mazelle and the James Taylor Quartet. He is also a talented chef and fencer . He is a manager at Ronnie Scott's jazz club.