The woman, who denied the murder allegations, was arrested after global police agency Interpol issued a red notice, the Korean National Police Agency said on Thursday.
She will now face extradition proceedings, they said.
“Police arrested the suspect at an apartment in Ulsan on Thursday following a stakeout with tips on her whereabouts and CCTV footage,” Seoul's National Police Agency said.
“The suspect is accused by the New Zealand police of having murdered two of her children, aged seven and 10 then, around 2018 in the Auckland area.
“She is been found to have arrived in South Korea after the crime and has been in hiding ever since.”
The woman told reporters she did not do it as she was escorted from a police station in the south-eastern city of Ulsan.
New Zealand has to formally seek extradition of the accused within 45 days for a South Korean court to review whether to send her back, the Ministry of Justice said.
New Zealand police launched a homicide inquiry in Auckland after the remains of the children were found by a family going through the contents of a storage locker they had purchased at an auction for abandoned goods near Auckland, New Zealand's biggest city.
The family who found the bodies were not connected to the deaths.
“Police plan to transfer the woman to the Seoul Central District Prosecutors Office in preparation for an extradition screening by the Seoul High Court,” the Yonhap News Agency reported.
Detective Inspector Tofilau Vaaelua in Auckland said the case was “a very challenging investigation”.
“To have someone in custody overseas within such a short period of time has all been down to the assistance of the Korean authorities and the co-ordination by our New Zealand Police Interpol staff,” the detective said.