The incident took place on Monday evening at Siddhirganj's Arambagh area, witnesses said.
Former Narayanganj unit Juba League vice president Selim Pathan and ward number 10 Chhatra League president Amir Hossain had a brawl over the area's control.
Their supporters soon fought each other with sticks and sharp weapons.
Pathan, hit by a bullet, and two others have been sent to Dhaka Medical College and Hospital.
Others have been admitted to local hospitals.
Siddhirganj police Inspector Md Rafiqul Islam said no complaints had been filed over the clashes.