Police made the stop at about 1:30 a.m. at South 11th Street in the city's South Ward.
The driver didn't get out of the vehicle but drove off and attempted to hit an officer, according to police.
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Officers then fired on the vehicle.
The suspect drove himself to University Hospital and walked into the emergency room with a gunshot wound, police said.
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Investigators examined the car with a hole in the driver's side window outside the hospital's trauma center.
The suspect's injuries are described as non life-threatening.
The Essex County Prosecutor's Office is investigating the incident.