"Engineers are now investigating the cause of the incident," said Air New Zealand's chief flight operations and safety officer, Captain David Morgan.
The plane has been taken out of service and towed into quarantine - off limits to the carrier's engineers - while Transport Accident Investigation Commission staff begin an inquiry.
Voice and data recordings have already been handed over, said the chief investigator of accidents, Captain Tim Burfoot.
"We'll focus on the actual aircraft itself in the first instance and see what we can find," he said.
"I can't recall something like this happening on an Air New Zealand flight before, but it does happen occasionally around the world.
"We need to find out what happened, before looking at why it happened. The answer might be simple, or it might take a while to reach, we really can't say at this stage."
If a major fault was uncovered, other carriers and manufacturer Boeing would be alerted immediately, Captain Burfoot said.
MP Claudette Hauiti was on the flight and tweeted that passengers were calm. But her oxygen mask failed to drop.
Air New Zealand refused to comment on that failure while the investigation is ongoing.
Aviation commentator Peter Clark said it was not uncommon for some oxygen masks to fail, but that "nobody was in any danger at all".
"The oxygen masks are really a stop-gap measure between altitude and getting to a breathable altitude," he said.
"The important message to come out of this incident is that people really need to listen and watch during the safety briefing so you know what to do when something happens."
Mr Clark said the problem might have come about while the aircraft descended and its pressurisation system malfunctioned, or there was a faulty valve or a leaky door.
Air New Zealand has 11 Boeing 737-300s, with an average age of over 15 years. They are being phased out.
Panic, sobbing, even a farewell text, after masks drop
Michael Donaldson said the pilot came out after landing. "He had that great line; he just stepped out and said, 'Well folks, that was the real deal.'
"There were a number of people who were hyperventilating ... and, after the incident especially, they needed a bag to breathe into.
"There was one woman nearby who was sobbing."
Olympic triathlete Hamish Carter, who was on the flight, said the experience was "a bit of a shock".
"It was relatively scary for a while, not something you'd expect to happen."
He noticed a definite change in cabin pressure when the oxygen masks dropped, with his ears popping. "I definitely felt something was a bit odd."
Passenger Thomas Wutzler texted a farewell note to his family.
"I wrote explaining to my family and friends that I loved them. We were descending very, very fast and it was shuddering quite badly."
MP Claudette Hauiti tweeted that passengers were calm, but her oxygen mask failed to drop.
Codey Jervis said he wasn't alarmed when the oxygen masks dropped. "It was annoying. It got in the way of me reading the newspaper."
