Arnesen will leave his job as Chelsea sporting director when his contract expires at the end of the season.
The former Denmark international has agreed a three-year contract with the German side.
“Hamburg is a special club with a long tradition and fantastic infrastructure,” said Arnesen in a statement.
“I am convinced that with the strength we have and the right decisions we can develop something big.”
Chelsea’s current chief scout Lee Congerto will follow Arnesen to Hamburg.
