The man was given interim name suppression when he appeared before Judge Rob Ronayne in the Manukau District Court on Friday.
Judge Ronayne rejected a request for bail and remanded the man in custody until next month.
A Counties Manukau police spokeswoman said the man "presented a weapon" and demanded cash.
However, a source told the Herald on Sunday a bank staff member claims to have had the machete held to the throat in the incident.
The police spokeswoman said the man left the bank on foot, but was followed by a member of the public. That person called police. The police Eagle helicopter followed the man until he was arrested, the spokeswoman said.
Both bank staff and the member of the public who called police won praise.
"Police commend the actions of bank staff who remained calm and co-operative during what was a very frightening situation.
"Police also wish to thank the member of public who followed the offender at a safe distance and was able to give a good description to both the Eagle helicopter and the attending police ground units."
An ANZ spokesman also celebrated the actions of bank staff.
"It was traumatic ... but they showed a lot of courage, remained calm and made sure a very dangerous situation didn't escalate into something worse. We're incredibly grateful they're okay."
A trauma support team visited the bank after the incident and staff had been offered counselling, the spokesman said.
He would not say how much cash was taken, except to say that it was "small". The money was later recovered. A review of security, standard after such an incident, is taking place, he said.
"The important thing is that staff or customers are not harmed. Our staff are trained to deal with such situations, and there is every indication that they followed their training to the letter in this case."
If convicted the charged man could be jailed for up to 14 years.
