The Dane has spent two years with Hamburg, including a spell as interim manager, but has failed to secure European football for a second successive season.
Arnesen was under contract at the imtech Arena until June 2014 but agreed to terminate that deal after a majority of the club's board failed to support an extension to his contract at a meeting on Tuesday evening.
A statement from the Hamburg board cited a "difference in opinions concerning the long-term direction of the club" for the decision to part with the 56-year-old.
The statement continued: "This season's performance is an indicator of his successful time here. We would like to wish him all the best for the future both personally and professionally."
Arnesen joined the German side in 2011 from Chelsea.