Born on 13 June 1923, Corbett was the son of Walter Corbett and Margaret Whitehead Corbett (née Robertson). He studied at the University of Otago, graduating Master of Science with first-class honours in chemistry in 1945. Corbett later undertook doctoral studies at the University of Cambridge under Alexander Todd on the structures of some of the metabolites—puberulic acid, puberulonic acid, and stipitatic acid—of Penicillium species. He showed that they contain the tropolone skeleton, and his PhD thesis, titled The structure of certain antibiotics was completed in 1950.
In 1949, Corbett married Enid Mary Carter in Cambridge, England, and they went on to have two children.