The men were eventually detained in Coventry after the drama started during a patrol in Blakenhall, Wolverhampton, at around 9.20pm last night.
Officers tried to pull over a BMW 1 Series which had cloned number plates in Upper Villiers Street but the hatchback rammed into them and fled.
Police lost track of the car but it was picked up by a police helicopter heading into Coventry about 20 minutes later.
The BMW was then tracked from the air as it drove at speed, dangerously overtaking cars and jumping red lights before eventually being abandoned in residential Eld Road in the north of the city.
Two men were then seen running into nearby Culworth Court but were found hiding in a block of flats by response officers who were guided in by the police helicopter.
The men, aged 29 and 23 and both from Coventry, were arrested on suspicion of car theft, dangerous driving and failing to stop for police.
Both men remained in police custody this morning for questioning.
The police car was damaged during the collision but no officers were injured.
The BMW 1 Series 118D M Sport was stolen in Coventry on September 24, West Midlands Police said.