After periods of post-doctoral research at the University of Newcastle upon Tyne and the University of Manchester in 1965 and 1966, Robinson was a faculty member in the Department of Chemistry at the University of Otago in Dunedin from 1967 to 2006. He became a professor in 1985 and the Mellor Professor of Chemistry in 1996, and served as the head of the Department of Chemistry from 1986 to 1998. When he retired in 2006 he was made a professor emeritus.
Building on his doctoral research, Robinson proposed that metal carbonyl clusters could participate in electron transfer reactions, and demonstrated that clusters are able to be reversibly reduced. Other research areas that he pursued included the design of organometallic compounds for use in biomedical research, and the development of molecular switches.