The shot man allegedly drove through Adelaide and eventually through Rundle Mall before police stopped the vehicle on Hindley St about 11.45pm.
As heavily armed Star Group officers surrounded the car, the man allegedly produced a gun and threatened the 3-year-old girl.
Police shot him and he was rushed to Royal Adelaide Hospital. His injuries are believed to be non-life threatening.
Dickson said police were initially unsure what vehicle the man had fled in.
He said just after 10pm police knew what vehicle the man was driving and began to track it with the PolAir helicopter.
Police tried to stop it by putting down road spikes at the intersection of Grand Junction and South Rds, but the car continued east until it turned onto Hampstead Rd at Northgate.
Dickson said the pursuit was difficult for police because they feared for the safety of the woman and children as the car drove through suburban Adelaide at "reasonably high speed".
Despite the road spikes, the car carried on down North East Rd into the city, where the driver tried to avoid police by driving up Rundle Mall.
Dickson said the car then turned north on King William St, onto North Tce then south onto Victoria St where it was halted by heavily armed Star Group officers.
"Once [officers] stopped the vehicle they attempted to arrest the offender," he said.
"When they were doing that, the offender threatened the 3-year-old girl with a firearm, and that is when a decision was made for the police officer to fire at the offender."
The woman and two children inside the car are not believed to have suffered serious physical injuries but were being treated for shock at Women's and Children's Hospital.
"Obviously there will be a whole lot of distress but we do not believe they have physical injuries," Dickson said.
"It is a terrible incident. You can't imagine how horrific it would have been for the young mother and her two children."
The shot man remains under police guard in hospital and is likely to face numerous charges.
