The plane has since been taken out of service and towed into quarantine - off limits to the carrier's engineers - while the Transport Accident Investigation Commission sent two investigators to begin an inquiry.
Its voice and data recordings has already been handed, chief investigator of accidents Captain Tim Burfoot said.
"We'll focus on the actual aircraft itself in the first instance and see what we can find," he told APNZ.
"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 any major faults are uncovered, which requires alerting other carriers around the world as well as the Boeing manufacturers, they would be alerted immediately, Captain Burfoot said.
MP Claudette Hauiti was on the flight, and tweeted that passengers were calm, but that 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 may have come about while the aircraft descended and its pressurisation system malfunctioned, or there was a faulty valve, or a leaky door.
"These are mechanical things - they happen. It's not something I would get stressed about," he said.
Air New Zealand has 11 Boeing 737-300s in its fleet, with an average age of over 15 years old.
The planes are progressively being phased out and replaced by new Airbus A320 planes which are bigger and more fuel efficient.
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 said.
He said 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."
Fellow passenger 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," he said.
"I think some people just found it unnerving. There was no one [who] got out of their seats, there wasn't chaos or panic but I think some people just deal with things like that in different ways."
