Border Guard Bangladesh (BGB) official Abdul Jalil told AFP news agency at least 12 bodies had been recovered after an all-night rescue operation, saying "they include 10 children, an elderly woman and a man".
At least 13 Rohingya, including three women and two children, were rescued after scouring the estuary of the Naf river, Jalil said.
Area coastguard commander Alauddin Nayan said the boat sank in the mouth of the Naf river near Shah Porir Dwip, on the southern tip of Bangladesh, with nearly 100 people on board.
"It capsized near Galachar (a coastal village in Bangladesh) with nearly 100 people," Nayan told AFP.
More than half a million Rohingya have emptied out of northern Rakhine and into Bangladesh since August 25, carrying stories of mass killings, gang rapes, and razing of whole villages.
Myanmar has denied allegations of ethnic cleansing, saying the military offensive was a "clearance operation" to flush out Rohingya fighters who had staged attacks on border posts in August.
More than six weeks after the violence erupted, Rohingya continue to arrive in Bangladesh.
Sunday's incident is the latest in a series of deadly boat disasters involving Rohingya refugees.
Most recently, on September 28, a boat carrying about 80 refugees overturned. Seventeen survived, while 23 were confirmed dead and the remainder declared missing.
Myanmar considers the Rohingya illegal immigrants from Bangladesh despite the ethnic minority living there for generations.