In a statement, police Supt Steve Kehoe said they were notified of the escape from the MIQ facility in Ellerslie just after 10:30am on Thursday.
COVID-19 Response Minister Chris Hipkins said he understood the person had been arrested.
"CCTV reviews are happening, and so on. I don't have all of the information yet.
"I do have some information that suggests someone hiding in a bush while a security guard walked past them.
"We haven't got confirmed information as to how exactly the person made it out of the facility, whether it was an area that was single-fenced or double-fenced.
"Anybody leaving an MIQ facility, particularly someone who is infected with COVID-19, is something that is utterly unacceptable."
Joint head of MIQ Brigadier Rose King said the escapee, a community COVID-19 case who entered the Novotel & Ibis Ellerslie facility on Wednesday evening, absconded in the early hours of Thursday.
"The fact that someone has absconded from one of our facilities is a disappointing and unacceptable breach," King said in a statement.
"Police and the Ministry of Health are leading the work on gaining an understanding of this person’s movements since they left the facility.
"We are investigating how this happened and will make more information available as we gather the facts."
Kehoe earlier said officers had cordoned off a property in Ōtāhuhu, dressed in full personal protective equipment, to arrest the escapee.
"All police staff that have responded to this incident have been following all health guidelines, in accordance with police COVID-19 procedures.
"We can assure our community every precaution is being taken."