The shooting happened just after 1 a.m. at the Starboard Restaurant in Dewey Beach.
Authorities say employees called to report a man with a gun.
When they arrived, officers talked with the man, who then took off running.
About an hour later, a Starboard Restaurant employee called police again to report the suspect had returned.
During a foot chase, an officer fired at the man, hitting him, according to reports.
The 21-year-old suspect later died at the hospital.
So far, no further details have been provided.