The cyclonic storm, which was formed by the depression in east central Bay of Bengal, hit the coast of Bangladesh between Cox's Bazaar and the port of Chittagong, the country's meteorological department said.
No casualties have so far been reported.
The authorities have moved hundreds of thousands of people to shelters. They had initially planned to make one million people leave their villages.
Mora struck the island of Saint Martin and Teknaf in the coastal Bangladeshi district of Cox's Bazar, where officials said on Tuesday about 200,000 people were evacuated to shelters.
"It is likely to move in a Northerly direction further and weaken gradually".
Significant damage was reported in refugee camps housing Rohingya Muslims from neighbouring Myanmar, BBC reported.
"Omar Farukh, a community leader in Kutupalong camp, said conditions were dire: 'Now we're in the open air.'".
"In some places, nearly every shanty home made of tin, bamboo and plastic has been flattened".
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Over 20,000 houses across various districts were devastated in the cyclone.
Numerous 22,000 people on Cox's Bazar's Moheshkahli Island, where 10,000 people died in a cyclone in 1991, moved to shelters.
Cyclone Mora is packing winds of up to 100 kilometres an hour (60 mph) and has prompted authorities to raise the weather warning to its highest level. Meanwhile all flights in and out of Chittagong Shah Amanat International Airport have been suspended till this afternoon.
Bangladesh is routinely hit by bad storms between April and December that cause deaths and widespread property damage.
Last year Cyclone Roanu hit coastal Bangladesh, leaving at least twenty four people dead. However, the administration has not yet released an exact account of the damage caused by Mora.
Medical units and volunteers were on alert in 19 coastal districts, said to Abu Syed Mohammad Hashim, an official at the Disaster Management Department.
Fishing boats and trawlers have been advised to remain in shelters. "Rain/thunder showers accompanied by temporary gusty/squally wind is likely to occur at most places over Mymensingh, Dhaka, Khulna, Barisal, Chittagong & Sylhet divisions and at many places over Rangpur & Rajshahi divisions with heavy to very heavy falls at places over the country".
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