It remains unclear which of the men was shot first in the altercation in the city's Nieder-Eschbach district, spokesperson from the state office of criminal investigation Udo Bühler said.
The man killed was allegedly on a rampage at the front door of his 44-year-old ex-wife, who alarmed police.
As two officers arrived at the scene in a patrol car, the man threatened them with a gun, police reported.
The officer underwent surgery for the life-threatening bullet wound.
The Hesse office for criminal investigation will take over the case.