She died at the scene about 8am, he said. The plane left for Fiji at 8:15am.
The woman had known medical history but didn't show any signs of ill health before boarding the plane, Mr Hunia said.
"Most of her family live [in New Zealand] so she comes here for a few months a year to see them. When we notified her next of kin, a family member said she had been in good spirits when they'd dropped her off this morning," he said.
Fiji Airways spokesman Shane Hussein said the flight continued without further incident.
"We're indeed saddened by the incident," he said.
The delay did not affect any other flights coming in or out of the airport, Auckland Airport Operations Centre duty supervisor Viren Fernandes said.
The case has been referred to the coroner.