Born in Montreal, Quebec, he attended Roslyn High School and Westmount High School in Montreal. He later studied at the University of California, Los Angeles and Pacific Coast University School of Law. Robson then found work in the prop department at 20th Century Fox studios. He eventually went to work at RKO Pictures where he began training as a film editor.

In 1940 he worked as an assistant to Robert Wise on the editing of Citizen Kane in addition to several other films. Both he and Wise benefited tremendously from producer and screenwriter Val Lewton, who promoted Robson from film editor to production assistant and later to director. In 1943, at the insistence of Lewton, Robson assisted Lewton and director Jacques Tourneur in a series of low-budget horror films produced by Val Lewton, including Cat People and I Walked with a Zombie. Later, Lewton was instrumental in promoting Robson to the director's chair for films such as The Seventh Victim (1943), Robson's first directing credit, and the troubled Isle of the Dead (1945).

His success at RKO lead to work on major film projects and in 1949 he was nominated for the Directors Guild of America Award for Outstanding Directorial Achievement in Motion Pictures for his work on the film noir drama Champion. That same year, he directed the popular romance My Foolish Heart and Home of the Brave, one of the first films to deal with the issue of racism. Robson briefly brought back his old mentor Val Lewton with fellow protégé Robert Wise in a partnership for film and television production, only to drop the ailing Lewton without explanation a few months later. Robson was nominated by the DGA again for the war drama The Bridges at Toko-Ri (1955), starring William Holden and Grace Kelly.

In 1958, Robson was nominated for an Academy Award for Directing for the major box office success Peyton Place and again the following year for directing Ingrid Bergman in The Inn of the Sixth Happiness. For these films he also received his third and fourth Directors Guild of America nominations. Robson also produced a number of films which he also directed including Von Ryan's Express in 1965. He directed 1967's Valley of the Dolls, a film panned by the critics but a success at the box office. In 1974 he directed Earthquake, the film that introduced "Sensurround".