Bangladesh has refused to let the two trawlers carrying about 500 people land on its territory despite UN calls to allow them in as a powerful storm bears down on the region.
Security officials said a small boat carrying about 40 people, including "starving" women and children, had come ashore.
Local people tipped off the coast guard and 29 people were detained and about 10 fled, police and security officials said.
Bangladesh officials would not confirm the Rohingya were from the boats stranded at sea.
But Shobbir Ahmed, a Rohingya at one of the refugee camps in southeastern Bangladesh's Cox's Bazar told AFP that two of his daughters who had been on the stranded boats were among those who landed in Bangladesh.
He said the women, aged 19 and 17, had boarded a trawler two months ago trying to get to Malaysia.
Ahmed said he paid a Rohingya intermediary about US$2,230 for them to make the trip.
Security officials, speaking on condition of anonymity, said the rescued Rohingya would be sent to the island of Bhashan Char in the Bay of Bengal where Bangladesh has built shelters for some 100,000 refugees.
The proposed relocation to the island in the cyclone-prone coastal belt has drawn criticism from rights groups and sparked protests by the Rohingya.
There are about one million Rohingya in camps in Bangladesh. Most fled a 2017 military crackdown in Myanmar and many favour going to Muslim-majority Malaysia where there are more work opportunities.
Bangladesh Foreign Minister A.K. Abdul Momen said last month that Bangladesh would not allow the Rohingya boats into its territory.
His declaration came after more than 60 Rohingya died on a boat that waited at sea for two months before it could land.
UN human rights commissioner Michelle Bachelet has made an urgent plea to Bangladesh to change its stance.
Thousands of refugees died in the Andaman Sea in 2015 on rickety fishing vessels that tried to reach Malaysia and Thailand. - AFP
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