Road spikes were used and the vehicle was abandoned in Sawyers Bay, resulting in the pair fleeing on foot and being tracked by dogs, found and apprehended.
Armed police were called in as a precaution, a police spokeswoman said.
An initial pursuit was called off because the manner of driving had become dangerous, she said.
Otago Daily Times staff witnessed a man in sports clothing, wet and muddy, being apprehended by armed police.
Police were first called about 2.20pm to Henley-Berwick Rd, southwest of Dunedin, after a motorist had been seen speeding.
Police signalled to the driver to stop, the driver failed to do so and there was a short pursuit.
They lost sight of the vehicle, but it was spotted again about 4pm.
"Police deployed road spikes, which safely brought the vehicle to a stop in Sawyers Bay," the police spokeswoman said.
"The two occupants of the vehicle then fled on foot.
"Police dogs were used to track the pair and they were located and arrested a short time later."
Senior Sergeant Anthony Bond, of Dunedin, said a 21-year-old man would appear in the Dunedin District Court tomorrow facing two charges of failing to stop and one of dangerous driving.
A female associate was arrested, but has been released.
- Additional reporting Tim Scott
