Born in Mexborough, West Riding of Yorkshire, Haigh studied drama at the Central School of Speech and Drama, then based at the Royal Albert Hall, London. He played the central role of Jimmy Porter in the premiere production of John Osborne's play Look Back in Anger in 1956 at the Royal Court Theatre. His performance in a 1958 Broadway theatre production of that play so moved one young woman in the audience that she mounted the stage and slapped him in mid-performance. For the film version released in 1959, he was passed over in favour of Richard Burton. Coincidentally, he went on to portray the Explorer and adventurer Richard Francis Burton in the BBC production of The Search for the Nile. He also briefly appeared in the Beatles' film A Hard Day's Night (1964).