The group, including 25 children, were taken to Sri Lanka's eastern port of Trincomalee, a navy spokesman said, adding that food and water had been provided.
"Medical checks have to be done before they are allowed to disembark," the spokesman said.
The mostly Muslim ethnic Rohingya are heavily persecuted in Myanmar and thousands risk their lives each year on long sea journeys, the majority heading southeast to Malaysia or Indonesia.
But fisherman spotted the drifting trawler off Sri Lanka's northern coast at Mullivaikkal at dawn on Thursday.
While unusual, it is not the first boat to head to Sri Lanka -- about 1,750 kilometres (1,100 miles) across open seas southwest of Myanmar.
The Sri Lankan navy rescued more than 100 Rohingya refugees in distress on a boat off their shores in December 2022.
The navy spokesman said Friday that language difficulties had made it hard to understand where the refugees had been intending to reach, suggesting that "recent cyclonic weather" may have pushed them off course.
Hundreds of thousands of Rohingya fled Myanmar for neighbouring Bangladesh in 2017 during a crackdown by the military that is now the subject of a United Nations genocide court case.
Myanmar's military seized power in a 2021 coup and a grinding war since then has forced millions to flee.
Last month, the UN warned Myanmar's Rakhine state -- the historic homeland of many Rohingya -- was heading towards famine, as brutal clashes squeeze commerce and agricultural production.
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