The engine on their small boat quit. They drifted out of the shipping lanes and after search aircraft failed to spot them, family members feared they were gone for good.
"Sometimes I thought, 'Oh God, are they gone or what?'" said Tonuia Filipo, the father of a survivor.
They were found 800 miles away. A fishing trawler spotted their 12-foot aluminum boat bobbing in the waters with the three boys waving frantically.
A Fiji Navy boat escorted them to port.
The boys, looking very shaky and sunburned, arrived in Fiji for medical treatment.
They survived on rainwater, two coconuts, the occasional fish they caught and a sea gull that landed on their boat.
The teenagers will be reunited with their families soon.
"We are so happy. Everyone is happy. All the Tokelau people are so happy," said Silifu Aluria, an aunt of one of the survivors.