The fight broke out on Wednesday night in Bayzid Bostami locality's Banglabazar area.
It was unclear what triggered the gun battle but locals say a dispute over land possession was the cause.
The injured have been admitted to the Chittagong Medical College and Hospital (CMCH), said the hospital's police outpost Sub-Inspector Pankaj Barua.
He said four youths, who sustained bullet injuries, were brought to CMCH around 10pm.
Doctors declared one of them, a man named 'Miraj', dead.
Barua identified the other three as Abdur Rahim, Nur Hossain and Anowar Hossain.
All the victims were activists of the ruling Awami League, locals say.