The powerful storm made landfall around 6am at Kutubdia, near the fishing port of Cox's Bazar, said the Met office.
Five hours after hitting the coast, Shamsuddin Ahmed, a director at the Bangladesh Meteorological Department, told bdnews24.com around 11:30am that the storm has started to weaken.
The disaster management ministry said the storm has damaged homes in Cox's Bazar town, St Martin’s Island, Kutubdia and Teknaf, but there is no report of any casualties.
The local administration, however, reported the death of an elderly man at a cyclone shelter.
“There have been strong winds coupled with heavy rains since the morning. The streets are deserted,” said Tahmidur Rahman, an official of the Cox's Bazar administration.
The maximum wind speed was recorded at 135km per hour in Teknaf, when the cyclone made landfall, according to Abdur Rahman, a senior meteorologist.
Cox's Bazar town saw winds of up to 114km per hour and Chittagong 128km.
All fishing boats and trawlers over the north Bay of Bengal and deep-sea were advised to remain in shelter as the sea still remains rough.
More than 317,000 people in six coastal districts have been evacuated by Monday night, said the disaster management ministry.
Cox’s Bazar district administration said some 538 shelters were ready to accommodate more than 500,000 people.
A total of 88 medical units, 6,010 local volunteers and 15,000 Red Crescent volunteers were on alert, while about 50,000 more volunteers were on standby in 19 coastal districts, according to Abu Syed Mohammad Hashim, director of the Disaster Management Department.
The disaster management and relief ministry has set up 24-hour control rooms in each of the upazilas in the districts that can be reached at 9540454, 9545115, 9549116 and 01715-180192.
About 18 million people live in the 19 coastal districts, with 10 of them being in high-risk areas.
The very warm waters of the Bay of Bengal caused a tropical low to develop into a depression on Sunday morning, which upgraded in to a deep depression by the afternoon.
The depression turned into a cyclonic storm by early Monday morning, and by evening the warning signal was upgraded to the highest level after it turned into a severe cyclonic storm -- equivalent to a strong tropical storm or Category 1 hurricane.
The storm's path suggests it might affect some northeastern Indian states.
The Indian Meteorological Department said the cyclone may weaken into a land depression by 6pm Tuesday. It has forecasted heavy rains in the states of Assam, Mizoram, Meghalaya and Tripura.