The trio -- a father, his son and a nephew -- were winched to safety by an Australian rescue helicopter late on Tuesday after managing to send mobile phone text messages to relatives with their approximate location, police said.
John Tabo, 38, his 20-year-old son John Jr and 16-year-old nephew Tom Tabo left their home on Murray Island in the Torres Strait between Australia and Papua New Guinea on April 17, bound for another island 70 kilometres away to fetch fellow members of a rugby team.
They were caught in heavy seas and were not heard from again.
Rescuers searched for the men and their five-metre boat, but gave up after five days because of an approaching cyclone.
The three were presumed dead until relatives received a text message from them on Tuesday saying their boat had drifted back near Murray Island.
"It was only from common sense that was used and especially for 21 days in conserving batteries for their mobile, and to find the right time to text was just a miracle," an island resident, Aven Noah, told ABC radio.
A visiting doctor who examined the men said they were dehydrated and had lost an average of 20 kilograms in weight, but should recover.
"They had managed to gather some rain water so they have been able to drink a little bit and at some stage they caught a squid and managed to eat that raw or at least two of them could stomach it," Jo Mclean told the radio.
Police said the three were incredibly lucky to have drifted back close enough to Murray Island to be able to use their phone.
They were found 17 nautical miles from their home island.