In a post on Twitter, the ministry said the three, including two children, had been caught trying to illegally enter the country.
Turkish authorities said on Monday that three New Zealanders, including a woman alleged to be a member of Islamic State, have been caught trying to enter Turkey illegally from Syria.
"They left New Zealand at the age of six, were resident in Australia from that time, became an Australian citizen, left from Australia to Syria, and travelled on an Australian passport", Ms Ardern said in Wellington.
Turkey's Government called the woman "S.A" and said she "identified as a DAESH (Islamic State) terrorist", but provided no further details.
'Any fair minded person would consider this person an Australian and that is my view too'.
Jacinda Ardern said the woman was an Australia-New Zealand dual national, but that Australia's Government recently stripped her of citizenship under national security laws.
'New Zealand, frankly, is exhausted of having Australia export its problems, ' she said.
Ardern said she has expressed her concerns to Australian Prime Minister Scott Morrison and that the two countries should be working more closely on these types of cases involving dual nationals.
"You can argue that our responsibility foremost is first to the children, then to the non-combatant and then lastly to the combatant".
Mr Morrison responded by saying his job was to protect "Australia's interests". 'They did not act in good faith, ' she said. "I think New Zealand, frankly, is exhausted of having Australia export its problems".
AAP's New Zealand correspondent Ben McKay quoted a NZ press gallery veteran and said it was "the angriest I've ever seen her".
He said he was scheduled to speak with Ardern today but added that legislation passed in parliament automatically cancels the citizenship of a dual citizen accused of being engaged in terrorist activities.
All New Zealanders have a right to come back to the country but that did not mean the country had to facilitate their return or pay for them to come home, he said. That's just not the right thing to do.
"If the shoe were on the other foot, we would take responsibility".
"The welfare of the children also needs to be at the forefront in this situation".
The Bulletin is The Spinoff's acclaimed daily digest of New Zealand's most important stories, delivered directly to your inbox each morning. "And I ask of Australia that they do the same".