The car drove "a few kilometres" to Princes St where police created a blockade so they could not reach the Octagon, Sgt Calvert said.
"Police took steps to prevent risk to the public."
The driver then turned off the car's lights and police abandoned the pursuit for public safety, he said.
The assailants crashed near Crawford St and lost tyres, before driving off again.
By that point the driver was manoeuvring the vehicle on three rims.
Officers located the car at the corner of Hanover and Grange Sts
A passenger from the vehicle threatened police and starting running from them.
He was apprehended by a police dog about 40m away from the car.
By that time less than 15 minutes had passed since the job was called in.
A 16-year-old male is being refereed to youth aid for a number of driving offences.
A 17-year-old male, who was treated for dog bite wounds at Dunedin Hospital this morning, will appear in the Dunedin District Court.