Similar measures have been taken in Myanmar however, some displaced people in Rakhine state have ignored calls for them to evacuate camps.
The UN said that more than 4.1 million people could be at risk from the cyclone, which started crossing Bangladesh’s low-lying coast on Thursday.
Mahasen first hit Khepupara on on the southern coast and has started bearing down on the ports of Chittagong and Cox’s Bazar.
Witnesses told the Reuters news agency that low-lying coastal areas were covered in waist-deep water, trees were uprooted and houses damaged.
Media reports said five people were killed, some by falling trees, and thousands of rickety huts were destroyed as the storm brought torrential rain.
Bangladesh evacuated more than 700,000 people living in the low-lying areas to thousands of cyclone shelters on Wednesday, while Myanmar announced plans to relocate about 166,000 people on its northwest coast.
The port in the Bangladeshi city of Chittagong and the airport in Cox’s Bazar were closed on Wednesday as the storm was likely to intensify and bring a storm surge of up to 2.1 metres, authorities said.
Our special correspondent, whom we are not naming because of reporting restrictions, said torrential rain and flooding was reported in Chittagong according to the Bangladesh Meteorological Department.
The correspondent said the government was alerting its citizens via community radio and text messages and had set up roughly 3,000 helters.
country is maintaining its storm warning to seven, on a scale with a maximum of 10.
On Tuesday, the storm killed at least seven people and displaced thousands in Sri Lanka as it tracked across the Bay of Bengal towards Bangladesh.
However, in Myanmar’s state of Rakhine, many Muslim Rohingya made homeless by communal bloodshed last year said they were too scared to move, reflecting their deep mistrust of the authorities and of local Buddhists.
“We could die here, we have no place to go,” said Yu Sut Taw, a Muslim man living in a camp on the outskirts of the state capital Sittwe, one of several in Rakhine which is home to a total of about 140,000 displaced persons.