"But it doesn't provide an open ticket for people from all the places that are impacted by climate change. It's still a very stringent test and it requires exceptional circumstances of a humanitarian nature."
The Tuvalu family moved to New Zealand in 2007, but has had no legal status in the country since 2009.
Their two children, aged 3 and 5, were born in New Zealand.
The father is a qualified teacher but has been a maintenance worker at a fast-food chain because he couldn't register as a teacher.
His several applications for work visas were refused.
In November 2012, the family lodged claims for refugee and protected persons status.
In March last year, their claims were dismissed and last month the tribunal turned down their appeals because they did not meet the refugee convention.
They successfully appealed against that decision on humanitarian grounds.
In a decision issued last month, the tribunal found "exceptional circumstances ... which would make it unjust and unduly harsh" for the family to return to Tuvalu.
Immigration lawyer Trevor Zohs, who represented the family with Carole Curtis, told the Herald on Sunday the effects of climate change should be recognised.
"A lot of people are affected by illness when they go back, they get sick from drinking polluted water. The island is porous so even when the water is not flooding, it penetrates the rocks under the land."
Appealing the decision on humanitarian grounds allowed discretion to be widened so the tribunal could consider whether it would be unjust to send a person back to their country. "Obviously their character, the fact they're going to contribute to New Zealand and that they have a connection to New Zealand is also taken into account," Zohs said.
The family has three generations of relatives living in New Zealand.
Kiribati man Ioane Teitiota's bid to become the world's first climate change refugee was rejected this year.
The 37-year-old moved to New Zealand with his wife in 2007 after deciding their life on the low-lying Kiribati island Tarawa was no longer sustainable because of rising seas.
He sought leave to appeal the tribunal's decision at the High Court, but that was dismissed.
In his ruling, Justice John Priestley said: "At a stroke, millions of people who are facing medium-term economic deprivation, or the immediate consequences of natural disasters or warfare would be entitled to protection under the Refugee Convention."