The third case of COVID-19 is a New Zealand resident in his forties from Auckland.
The man's family members returned to the country after visiting Iran, but he did not travel with them.
Director-General of Health Dr Ashley Bloomfield confirmed in a media briefing on Thursday that it was likely a case of "family transmission".
"There is what appears to be a clear link with travel to Iran by a close family member. Our view remains that with continued vigilance the chance of widespread community outbreak remains low," he said.
Results of the test were formally reported to the Ministry at about 6pm on Wednesday evening.
The man is currently in self-isolation at home with clinical support from public health and district health board staff.
Neither the man or his family requires hospital-level care.
A case of 'family transmission'
"We have one case that's confirmed and the person who travelled back from Iran is a probable case," Dr Bloomfield said.
"The implication is that they [the probable case] had been infected with coronavirus, developed a mild illness and there has been transmission in the family to the person who is the third case.
"That person developed a mild illness after they got back to New Zealand. They are now asymptomatic, however a close family member has had an illness that tested positive as COVID-19.
"There's no indication to test [the wider family]. The only other family member that has been tested is the case's partner because she still has some mild symptoms. We test people who have symptoms."
The family members travelled via Qatar Airways on Flight QR0920 from Doha to Auckland, which departed February 22 and arrived on February 23. Healthline is aware of the seat numbers and is able to advise any passengers if they are considered a close or casual contact.
Auckland Grammar and Ormiston Junior College have been notified about the positive test as there is a family member at each of the schools. The students are at home in isolation and have no symptoms.
"These are Kiwi families affected by a virus that is part of a worldwide outbreak. What they need is support and understanding. Our task is to ensure they have all the support and healthcare they need," Dr Bloomfield said.
Seven possible cases are currently under investigation.