One passenger was found dead while many of the 138 plucked from the boat were dehydrated, said navy spokesman Kosala Warnakulasuriya, adding that the vessel had been adrift for 10 days before it sank on Sunday.
"We sent three ships for the rescue at a location 50 miles off the eastern coast of Akkaraipattu," Warnakulasuriya told AFP. "Some have been admitted to a local hospital."
Police said 14 were Myanmar nationals while the others were Bangladeshis.
Fifteen of the survivors, including two women and two children, were hospitalised with acute dehydration, police said in a statement.
"We have difficulty in communicating with the survivors so we have asked the two embassies to send us translators," police spokesman Prishantha Jayakody told AFP.
He said statements of survivors would be recorded and they would be moved to a temporary shelter in Colombo under judicial supervision. In the meantime, the authorities at Oluvil fishing harbour were giving them shelter.
"We are certain that they were not trying to enter Sri Lanka, but their boat developed trouble in mid-sea and they drifted close to our shores," Jayakody said.
The early-morning rescue came amid stepped up naval patrols to deter Sri Lankan fishing boats from taking would-be illegal immigrants to Australia.
Authorities arrested more than 1,200 people trying to leave the island illegally last year. Many of those who make the perilous journey pay up to $3,000 for a place on trawlers run by people-smugglers.
Warnakulasuriya said the passengers rescued on Sunday identified themselves as Bangladeshi and Myanmar nationals but it was not yet known where they came from or were headed.
Reports from local fishermen alerted fishing authorities who in turn asked for help from the navy which mounted a 20-hour search and rescue operation, officials said.
They said it was unclear if those identified as Myanmar nationals were Rohingya - members of a stateless Muslim minority described by the UN as one of the world's most persecuted groups - who had fled Myanmar.
An explosion of tensions between Buddhist and Muslim communities in Myanmar's western state of Rakhine since June 2012 has triggered an seaborne exodus of Rohingya.
Thailand's navy blocked more than 200 Rohingya boat people from entering the kingdom late last month as part of a new policy, under which they will be given food and water but barred from landing if their boat is seaworthy.
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