The driver finally pulled over three blocks down the road on Bond Street at about 2 a.m.
Police said one officer questioned the driver from the passenger's side.
"We believe that when he reached into the vehicle, the driver accelerated, taking the officer with him inside the passenger side," said Sgt. Rick Martinez, Anaheim Police Department.
The officer struggled with the driver for about a block before firing his gun, hitting the driver in the head.
The minivan came to a stop when it crashed into a parked vehicle on Elm Street. The driver was taken to UCI Medical Center in critical condition.
The officer who fired the shot is a veteran on the force. He was treated for minor injuries and released.
The name of the suspect has not been released, but police said he is a 21-year-old parolee wanted for narcotics violations.
"We do believe that drugs were involved in this," Martinez said.
It's unclear what type of traffic stop the officers were pulling the minivan over for.
The driver was in grave condition on Friday afternoon.