Armed Offenders Squad members supported other police staff when they went to the property, but the man was not there.
"At about the same time, police received a report of an individual at a nearby address who matched the description of the person police were trying to locate."
A police dog handler began tracking him through several properties, where he pointed a gun at one woman.
He emerged from a property on Plymouth St. When challenged by police, he pointed a handgun at officers.
"Two shots were fired at the offender. We now believe he sustained injuries to his leg and his head from these two shots," Ms Kura said.
Police gave immediate first aid at the scene.
A second person has also been arrested.
She said police were in the "very early stages of a significant investigation into this matter" as staff from other districts arrived to help with the investigation.
The Independent Police Conduct Authority has also been notified.
"I'm satisfied this incident was managed professionally and appropriately by the staff involved," Ms Kura said.
She acknowledged the "shock of the community", but said it was an "isolated incident" and there was no reason to believe there is any risk to the wider community.
A Bell Street resident told NZME she went out into her yard about 8.30am and saw a man standing in her garden, with her dog barking at him.
"I think he was talking to our dog or trying to get our dog to shut up," the woman said.
The man did not see her, and she went inside to call police. She said they were there "within seconds".
Nearby Keith Street resident Gordon McMeeken spotted a police officer jumping over his fence with a police dog.
"I opened the door and said 'can I help?"' Mr McMeeken said.
The officer told him to go back inside as he would put the dog off the scent.
The man also ran through another property on Keith St, near Plymouth St.
The woman, who didn't want to be named, was heading inside after initially investigating the police presence, but then turned around and saw her back gate was open. She went over to a police officer and asked if he knew why.
"He says, 'he [the offender] ran through your property and we jumped after him'."