"He even promised to expedite the processing and assessment of our visa application, and said Zara will be issued with a dependent child visa once it is approved."
The family were also told that they did not have to pay the application fees for Zara.
"I guess this is their way of saying sorry about the letter," Lambojo said.
INZ confirmed that Elms had called Zara's mother "to apologise for the tone and nature of the correspondence the family received about her visa status and to explain the immigration process generally".
An INZ spokesman said: "Peter has confirmed that record of the telephone conversation is correct with the proviso that he was unable to give any guarantee about the outcome of Mrs Lambojo's visa application."
A devout Christian, Aileen said she "cried tears of joy" after the phone call, and one of the first things she did was give thanks to God and called her pastor at Christ New Creation International to share the good news.
"I am just so thankful that there is some stability now, and that our family won't be broken up," she said.
Both Aileen and her husband Arnold were on interim visas when Zara was born, which meant she was deemed to have the same status as her parents.
They have current applications with INZ and are awaiting a decision.
A law change in 2006 meant babies born in New Zealand no longer had automatic rights to citizenship.
A request for a visa for Zara was lodged after her interim visa had expired, which by law, made her unlawful in New Zealand.
In a letter addressed to the child declining the visa request, INZ said: "You are now unlawfully in NZ and must leave NZ immediately. If you do not leave NZ voluntarily you will be liable for deportation."
Elms said the agency would be reviewing the standard letters about deportation liability for children under 18 as a matter of priority.
He said it would also be ensuring communications are sent to parents and guardians.
"We will have the wording of the revised letters finalised and in use by next week," Elms said.
