Three young children from the same family were among those killed in the disaster, which comes just weeks after a cyclone battered the area, destroying camps housing thousands of Rohingya refugees.
Police warned that the death toll would likely rise as emergency workers reached remote parts of the Chittagong Hills, where telephone and transport links had been cut.
"The recovery work is still going on. The death toll could rise as many areas still remained cut off," the head of the Department of Disaster Management Reaz Ahmed told AFP.
Many of the victims were from poor tribal communities in the remote hill district of Rangamati, close to the Indian border, where 48 people were killed when mudslides buried their homes.
"Some of them were sleeping in their houses on hillsides when the landslides occurred," district police chief Sayed Tariqul Hasan told AFP.
"We've evacuated some of the affected villagers to schools and other safe places."
Police also ordered the evacuation of thousands of people living in slums at the base of hills in the neighbouring district of Chittagong, where 23 people have been confirmed dead.
At least 126 people were killed in that district when a massive landslide buried a village a decade ago.
Six were killed in the nearby district of Bandarban, among them three children who were buried by a landslide as they slept in their home.
Ahmed said disaster response teams had been deployed, but had not yet been able to reach all the affected areas.
The monsoon rains came two weeks after Cyclone Mora smashed into Bangladesh's southeast, killing at least eight people and damaging tens of thousands of homes.
Rohingya leader Mohammad Anam said the latest rains had further worsened conditions in makeshift camps that were badly hit by the cyclone.
"We're living in constant fear of landslides," he told AFP.
Around 300,000 Rohingya, stateless mainly Muslim ethnic minority, are living in camps in southeastern Bangladesh after fleeing persecution in Myanmar.
Heavy monsoon rains also pounded the capital Dhaka and the port city of Chittagong in the district of the same name, which experienced 222 millimetres of rain, disrupting traffic for hours.
A ferry sank in the river Buriganga in Dhaka on Monday evening with an estimated 100 passengers aboard, police said, adding all the passengers had managed to swim ashore.
Among the victims in Rangamati district were at least four soldiers who had been sent to clear roads after an earlier landslide.
Thousands of troops are stationed in Rangamati, where a tribal insurgency raged for two decades, and which still suffers sporadic violence.
"The soldiers were sent to clear roads hit by landslide in Manikchhari town when they were themselves buried by a second landslide," armed forces spokesman Lieutenant Colonel Rashidul Hassan told AFP.
"Our team is working here to clear the dirt. We'll know the full extent of the damage and casualties after they complete the work."
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