Locals informed the coastguard and 29 people were arrested while around 10 people fled, police and security officials said.
Those rescued will be sent to the island of Bhashan Char in the Bay of Bengal where Bangladesh has built shelters for some 100,000 refugees, according to officials speaking on condition of anonymity.
The island proposed for relocation is located in a cyclone-prone area, which has drawn criticism from rights groups and protests by the Rohingyas.
The Director of the Arakan Project monitoring group Chris Lewa, an organization working to improve the situation of the Rohingyas said those that landed had likely arrived on a small boat from one of the vessels stranded at sea, with hundreds of women, men, and children on board.
Two trawlers were stuck in the Bay of Bengal after Malaysia, which has imposed restrictions on all boats because of the COVID-19 pandemic rejected them, while Thailand also indicated it will refuse them entry.
As a strong storm is moving toward the region, the United Nations (U.N.) along with rights groups have been urging Bangladesh to let the boats disembark on its territory, but the country has refused.
Bangladesh Foreign Minister AK Abdul Momen said last month that Rohingya refugees are "not Bangladesh's responsibility."
His comments came after more than 60 Rohingyas died on a boat that waited at sea for two months before it could dock.
For years, Rohingyas have fled Myanmar and Bangladesh by boat for Thailand and Malaysia when the seas are calm between October and April. Hundreds died in 2015 after a crackdown in Thailand led smugglers to abandon their human cargo at sea.
Almost one million Rohingya refugees currently live in overcrowded camps in Bangladesh. Since 2017, they have been fleeing Myanmar to escape genocide.
An estimated 600,000 remain in the Southeast Asian country.
Deprived of their basic rights and confined to camps or villages without freedom of movement, they are regularly subjected to government persecution and brutality.
The Rohingyas are currently "one of the most persecuted minorities in the world," according to the U.N.