A woman and her two children were among those killed in a mudslide at Betbunia in the southeast. Weather officials said nearly 225 mm (9 inches) of rain fell overnight in Chittagong port city, severing road links with the Chittagong Hill Tracts further to the southeast.
The storm originating in the Bay of Bengal made landfall around Monday midnight, a weather official said, adding more rains were forecast across the country over the next couple of days.
At least 20 fishing boats were reported missing at sea, police said.
Streets in Chittagong were under knee-deep water, forcing authorities to shut down offices and schools, a resident said by telephone from the city.
Normal life was also disrupted at Cox's Bazar, the country's main tourist resort, following the rainstorm.