Hawking, who has motor neurone disease, completed his last day as Lucasian Professor of Mathematics on September 30.
The university said Professor Green had been elected as the 18th person to take up the position.
Green will start work in his new role on November 1. Earlier scientists who held the Lucasian professorship included Sir Isaac Newton.
Green already holds the John Humphrey Plummer Professorship of Theoretical Physics at the university. He is a pioneer of string theory, the idea that the fundamental building blocks of space and time are tiny vibrating strings.
Peter Haynes, head of the department of applied mathematics and theoretical physics, said: “Michael Green has played a leading role in theoretical physics research in the department since 1993.”
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