Authorities say they evacuated nearly 600,000 people from vulnerable areas before the storm hit the coastal district.
Cyclone Mora battered Bangladesh on Tuesday, killing at least five people, damaging thousands of homes and ripping through a camp housing thousands of Rohingya refugees who had fled violence in Myanmar.
Authorities said they evacuated nearly 600,000 people from vulnerable areas before the storm hit the coastal district of Cox's Bazar early Tuesday, bringing winds of up to 135 kilometres (84 miles) per hour.
Disaster management authorities said five people had been killed, four of them crushed by falling trees in the area. It was not immediately clear how the fifth person died.
Authorities in Cox's Bazar said at least 17,000 homes, excluding the Rohingya shelters, had been damaged in the district, with many low-lying villages inundated by a storm surge reaching four feet (1.3 metres).
Some of the worst damage was at the camps housing the 300,000 Rohingya refugees living in Cox's Bazar, many of them in flimsy huts.
The local head of the International Organisation for Migration, which coordinates relief for the refugees, said the bulk of the homes at one camp had their roofs blown off.
"We're already in the field. At Kutupalong camp, which I am visiting now, some 60-70 percent of the plastic roofs have been blown away. Some mud walls have collapsed," Sanjukta Sahany said.
"Rohingya people have already started repairing their houses."
Kutupalong houses nearly 14,000 registered refugees, although many more recent arrivals who lack official refugee status are also said to be living there.
Community leaders said there had been no attempt to evacuate undocumented Rohingya, although those with official refugee status were alerted about the dangers.
Abdul Salam, a Rohingya community leader, said around 20,000 homes had been damaged and some residents injured.
"In some places, almost every shanty home made of tin, bamboo and plastics has been flattened," he added.
Cox's Bazar has for years been home to hundreds of thousands of Rohingya, a stateless minority living mostly in Myanmar.
Their numbers have swelled since a brutal crackdown last October by the Myanmar military sent 70,000 fleeing across the border into Bangladesh.
Away from the camps, authorities had evacuated nearly 600,000 people to cyclone shelters, schools and offices after raising the highest number 10 weather danger alert.
TRT World's Sarah Jones reports.
Cyclone Mora comes days after heavy rains in Sri Lanka killed at least 193 people, many of them buried under landslides, and brought the worst flooding the island has seen in 14 years.
South Asia is frequently hit by flooding in the summer with the arrival of the annual monsoon rains.
Mohammad Anam, a Rohingya who fled to Bangladesh last year after an outbreak of violence in Myanmar, said there had been no attempt to evacuate the minority community, many of whom are undocumented.
"Nobody came to alert or evacuate us. When the storm came we rushed to local schools to take shelter," he told AFP.
However Abul Hashim, a spokesman for the disaster management department, said the authorities only evacuated the most vulnerable from low-lying coastal areas.
"Rohingya live in hilly areas. There is no chance for these areas to be inundated by storm surge," he said.
Authorities called in all fishing vessels and suspended flights to and from airports in Chittagong and Cox's Bazar.
The weather department said the cyclone lost power as it headed into the Rangamati hill district and turned into a depression, bringing rain.
Bangladesh is routinely hit between April and December by severe storms that cause deaths and widespread damage.
In May last year Cyclone Roanu hit the southern coast of Bangladesh, leaving 20 people dead and forcing half a million to flee their homes.
In 2007 Cyclone Sidr killed nearly 4,000 people and caused damage worth billions of dollars.
In neighbouring India, the National Disaster Management Authority advised fishermen in the eastern coastal state of West Bengal bordering Bangladesh to stay in port.
Weather authorities warned of strong winds and heavy rain across much of northeast India, although the cyclone was expected to weaken as it crossed the country.