Most of them were killed by falling trees in the districts of Cox's Bazar and Rangamati, officials told the BBC.
Hundreds of houses were fully or partly damaged, the officials said. Significant damage is reported in refugee camps housing Rohingya Muslims from neighbouring Myanmar.
The authorities have moved hundreds of thousands of people to shelters.
Cyclone Mora made landfall at 06:00 local time (00:00 GMT) between the fishing port of Cox's Bazar and the city of Chittagong, with winds of up to 117 km/h (73mph), the country's Meteorological Department said.
It had previously warned that low-lying areas of Cox's Bazar, Chittagong and many other coastal districts were "likely to be inundated" by a storm surge of 1.2m-1.5m (4-5ft) above normal levels.
People have been evacuated to shelters, schools and government offices.
Fishing boats and trawlers have been advised to remain in shelters. Flights in the area have been cancelled.
About 20,000 houses in refugee camps for Rohingya were damaged, community leader Abdus Salam said told AFP news agency.
"In some places, almost every shanty home made of tin, bamboo and plastic has been flattened. Some people were injured, but no-one is dead," Mr Salam added.
Large camps have been set up in Cox's Bazar for hundreds of thousands of ethnic Rohingya who have fled violence in Myanmar.
Bangladeshi officials told the BBC that a clear picture was still not available due to poor communication with many affected areas.
A number of houses were also damaged in western Myanmar.
Tracking website Tropical Storm Risk forecasts that Cyclone Mora will move northwards past Chittagong, weakening as it moves further inland and downgrading from a Category One hurricane to a tropical storm.
Parts of eastern India are expected to be affected later on.
The Bay of Bengal is prone to storms, and Bangladesh is often hit by severe weather during the monsoon season from the middle to the end of the year.
Last year Cyclone Roanu hit coastal Bangladesh, leaving at least 24 people dead.