People were rescued from balconies at the rear of the property, a Fenz spokesman said.
Ten fire trucks rushed to the scene and contained the fire in less than an hour, the spokesman said.
There were no injuries reported.
Two fire investigators are looking into the cause of the blaze.
Kāinga Ora’s regional director for Counties Manukau, Angela Pearce, told the Herald the agency was “very concerned to learn about a fire at one of our blocks of flats in Auckland last night”.
There are 56 flats in the complex, Pearce said.
Water and electricity are not working in the building due to the fire and the battle to put it out, she said.
“Our team is on-site continuing to assess the impact of the fire and supporting customers.”
Six residents had been moved from their homes after the fire, and Kāinga Ora was arranging alternative accommodation for them while the building was assessed.
“Our immediate focus was on checking to make sure all our customers were safe. Thankfully, no one was injured,” she said.
“We are arranging alternative accommodation for these customers while we assess the extent of the damage to their homes and organise any repair work that is necessary.”
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