The man who was standing outside the Zig Zag cafe at 1.30am on Sunday morning said he saw the silver BMW X5, pictured, pursued slowly by a marked police car with its lights on.
He said he saw an officer dressed in black standing in front of the car and shoot.
The witness, who said the whole incident was over within ten seconds, said he heard someone shouting before the shots were fired.
"There was two officers definitely with guns getting down from the car," said the man, who was facing towards Hessle Road.
"I didn't hear exactly - the music was playing - I heard boom boom, just two shots. It all happened in 10 seconds.
"Definitely the officer was very qualified, they were not normal officers.
"When the officers were coming from the car they had guns."
Within two or three minutes around 10 other police cars showed up, the witness said.
A police cordon remains in place on Hessle Road, with traffic being diverted.
White-suited forensics officers were working around a blue tent erected in the middle of the road.
Two traditional police helmets have also been left in the middle of the road, highlighted with yellow forensic markers, as well as paramedics' equipment.
Local residents expressed shock. Martin Alred said: "There's not a lot of crime, you have the prostitutes which you get used to.
"For someone to be shot is very shocking, especially at this time of year."
A shopworker nearby said: "Something like this is a very rare occurence - you have your knife crime, but you don't hear about firearms.
"I've heard the person involved has a firearm - all sorts."
At a press conference on Sunday Deputy Chief Constable Chris Rowley said officers had to make “very difficult decisions in very difficult circumstances”.
The officers had been deployed to reports of a man with a gun.
The man is in critical but stable condition in hospital. No one else was injured.
DCC Rowley said: “Officers were deployed to the area after a report was received that a man was in the street and believed to be in possession of a firearm.
“A mandatory referral has been made to the Independent Office of Police Conduct.
“We are still in the very early stages of the investigation, however we do not believe this incident to have any terrorism connection or links.
“Our thoughts are with the family of the man and also with those officers who were involved in the incident.
“In incidents like this our officers have to make very difficult decisions in very difficult circumstances.”
Anyone with information is asked to ring 101.