Robinson was born in London, and attended Winchester College and Corpus Christi College, Oxford. He joined the staff of the Victoria and Albert Museum in 1939 and was shortly afterwards transferred to the Department of Metalwork where, apart from war service in the East from 1939 to 1946, he spent the rest of his career. He served as Keeper of the Department of Metalwork from 1966 until his retirement in 1972. In 1967 he was elected honorary president of the To-ken Society of Great Britain and was president of the Royal Asiatic Society from 1970 to 1973. He was a Fellow of the Society of Antiquaries and a Fellow of the British Academy.
His principal fields of scholarship were: Persian miniature paintings (he developed the standard classification and chronology), Japanese swords, and the artist Utagawa Kuniyoshi (he established the artist's place among the great masters of ukiyo-e).
He died in 2005. He had married twice: firstly Mary Stewart, who died in 1954, and secondly Oriel Steel, with whom he had a son and a daughter.