The Independent Police Conduct Authority oversaw a police investigation looking into Stacey Owen Waaka's death.
Officers acted in accordance with their duties and responsibilities during the arrest, using minimal force, the authority said.
Waaka had been arrested after assaulting a police officer and running away.
During the pursuit, an officer discharged his Taser twice but the barbs did not hit Waaka.
Shortly after being arrested, the 39 year old became unresponsive and died at the scene despite the officers' attempts to revive him.
The investigation found Waaka suffered a cardiac arrest, which was attributed to drug toxicity in combination with physical exertion.
It found that the officers used minimal force during the arrest, placing Waaka in handcuffs and escorting him to the patrol car, and responding appropriately upon finding him unresponsive.
The coroner will make a formal ruling on the cause of death.