They were vaccinated with the same unsterilised syringe, and the vaccine was not stored properly, according to an investigation into the deaths.
As a result, the immunisation campaign caused severe infection and sepsis.
The team administering the vaccinations was “neither qualified nor trained” to do so, the health minister said.
About 300 people were vaccinated during the campaign in the Kapoeta region, including 32 other children who fell ill, but survived, Riek Gai Kok added.
But a report from the Associated Press said that in this case, children as young as 12 were administering the vaccine to others.
Sepsis is a potentially deadly immune response triggered by an infection which spreads quickly in the body.
It can lead to multiple organ failure and death.
South Sudan declared independence in 2011, but has struggled to maintain basic services – including healthcare – after a civil war erupted in 2013.
The United Nations estimates more than one million children have fled the country during the conflict, and a million more are displaced within the country.