Thousands hunkered down in monasteries, pagodas and schools as Cyclone Mocha made landfall on Sunday afternoon.
The category-five storm tore rooves off buildings with wind speeds of up to 130 miles per hour (209km/h).
More than 500 bamboo shelters in Cox’s Bazar, the world’s largest refugee camp, have been destroyed.
Landslides and floods hit the area while high winds crumpled mobile phone towers, cutting off communication.
Myanmar’s military information office said the storm had damaged houses, electrical transformers, boats and lampposts in Sittwe, Kyaukpyu and Gwa townships.
It also tore rooves off of sport buildings on the Coco Islands, about 264 miles south west of the country’s largest city, Yangon.
More than 4,000 of Sittwe’s 300,000 residents were evacuated to other cities and more than 20,000 people are sheltering in sturdy buildings such as monasteries, pagodas and schools located on the city’s highlands, said volunteer Tin Nyein Oo.
Lin Lin, chairman of a local charitable foundation, said there was not enough food in the shelters in Sittwe after more people arrived than expected.
Authorities say more than 500,000 people were evacuated from their homes in south eastern Bangladesh.
Many arrived at shelters in rickshaws and on foot, bringing cattle, chickens, goats and mats to sleep on.
Sumi Akter, who lives on a riverbank, told the BBC: “I didn’t want to leave my house.
“I wish the homes we lived in were built more strongly.”
Titon Mitra, the UN Development Program representative in Myanmar, tweeted: “Mocha has made landfall. 2m people at risk. Damage and losses are expected to be extensive. We are ready to respond and will need unhindered access to all affected communities.”
Myanmar state television reported that the military government is preparing to send food, medicine and medical personnel to the storm-hit area.
A rescue team from the country’s eastern Shan state announced on its Facebook social media page that they had recovered the bodies of a couple who were buried when a landslide caused by heavy rain hit their house in Tachileik township.
Local media reported that a man was crushed to death when a banyan tree fell on him in Pyin Oo Lwin township in the central Mandalay Region.
In May 2008, Cyclone Nargis hit Myanmar with a storm surge that devastated populated areas around the Irrawaddy River Delta. At least 138,000 people died and tens of thousands of homes and other buildings were washed away.