The trio - Samuel Pelesa and Filo Filo, both 15, and Edward Nasau, 14 - told rescuers they survived on rainwater they collected, a handful of coconuts, raw fish and a seagull that landed on their 12-foot- (3.5-meter-) long aluminum boat.
The boys set off Oct. 5 from their home island to one nearby.
It's not known how they went missing, but the outboard motor may have broken down at sea.
Worried family members reported them missing and the New Zealand air force launched a sea search. No sign of the tiny boat was found, and the village of 500 people held memorial services, expecting never to see the boys again.
They were picked up Wednesday by a fishing trawler, undernourished, severely dehydrated and badly sunburned, but otherwise well. The ship's first mate said the area they were in is way off any normal commercial shipping routes.
They drifted 800 miles (1,300 kilometers) from where they set out - Tokelau, a bucolic collection of coral atolls north of Samoa that is New Zealand's territory.
A Fiji navy patrol boat met the trawler Friday and escorted it into the harbor of its capital, Suva. The teens were met by New Zealand consular officials and taken directly to a hospital for medical checks. Looking thin, the three walked off the boat without speaking to reporters.
Tai Fredricsen, first mate aboard the tuna boat San Nikuna, said a crew member spotted a small vessel bobbing in the open sea northeast of Fiji on Wednesday. "We knew it was a little weird," he said.
As it edged closer to investigate, the crew saw three people aboard waving frantically and asked them if they needed help.
"All they could say was 'thank you very much for stopping,"' Fredricsen told New Zealand's National Radio. "In a physical sense, they look very physically depleted, but mentally - very high."
After the rescue, one of the boys managed to reach his grandmother by phone from the fishing boat. As news of their survival spread, the village erupted in joy.
"It's a miracle, it's a miracle," said Tanu Filo, the father of Filo Filo. "The whole village, they were so excited and cried and they sang songs and were hugging each other in the road. Everybody was yelling and shouting the good news," he told Radio New Zealand International.
Fredricsen said the boys reported having just two coconuts with them when they set out. During their ordeal, they drank rainwater that collected in the boat and ate fish they had caught. Once, they managed to grab a bird that landed on the boat and they devoured that, Fredricsen said.
The rescue came not a moment too soon: Fredricsen said they had begun to drink sea water because it hadn't rained in the past few nights.
He said the tuna boat's crew had given the boys small portions of fruit and fluids.
Cmdr. Francis Kean, Fiji's naval commander who was among those who met the teens, said they had been unable to keep down solid food. The boys would be fed fluids and carefully watched by doctors at a military hospital.
"They were surviving on rainwater, sea water, bird meat and flying fish, so that's kept them alive," Kean told reporters. "They suffered from severe dehydration, as you notice when they got off some of them were still weak on their legs."
Kean said the teens would not be available to speak to the media until they were healthier.
Fredricsen said the waters where the teenagers were spotted are isolated and commercial vessels rarely pass through. The San Nikuna was there trying to shorten its return journey to New Zealand.
The boys come from the atoll of Atafu, one of three that comprises the tiny Tokelau island group where 1,500 people live.
Atafu, Nukunonu and Fakaofo, picture-perfect South Pacific islets, lie 300 miles (500 kilometers) north of Samoa, surrounded by 128 mostly uninhabited coconut palm-covered islets. The territory has a total land area of just 4.7 square miles (12.2 square kilometers).