Cyclone Mocha is bringing heavy rain and winds of up to 195kph (120mph), which could see dangerous flooding on land around the Bay of Bengal.
Storm surges of up to four meters could swamp villages in low lying areas.
There are fears it may hit the world's largest refugee camp, Cox's Bazar, where nearly one million people live.
Forecasters warned Cyclone Mocha could be the most powerful storm seen in Bangladesh in nearly two decades and around 500,000 people have been evacuated to safer areas.
Heavy rain and driving winds began battering the region as the storm system moved to the coast, with reports of increasing disruption near the city of Sittwe, in Myanmar, from about 13:00 (07:00 GMT).
Electricity and Wi-Fi connections were disrupted across much of the Sittwe area, where videos show the tide rising dramatically and carrying debris down flooded streets.
Rescue teams in Sittwe said they were receiving distress calls from people trapped by flooding, but that it was still too dangerous for them to provide help.
A telecom tower in was brought down by the high winds as the cyclone approached. Videos shared on social media also showed roofs being blown off houses and billboards flying off buildings in Yangon amid teeming rain.
Local media reported that a 14-year-old boy was killed by a falling tree in Myanmar's Rakhain State.
In Bangladesh too gale force winds started to blow, accompanied by heavy rains. In Cox's Bazar, the streets emptied out as the storm started to pick up, with trees swaying in the wind and skies getting darker.
Many shelters set up in schools and monasteries in Cox's Bazar are now full and overcrowded. Police in the area spent much of Saturday warning people to take cover and avoid the beach area. The government also issued an official landslide warning linked to the predicted very heavy rainfall forecast for Sunday,
The Bangladeshi meteorological department office said the maximum sustained wind speed within around 75km (45 miles) of the center of the cyclone was around 195 kilometers (120 miles) per hour with gusts and squalls of 215 kilometers per hour.
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