Cyclone Bulbul, expected to unleash surges as high as 7ft, has prompted the evacuation of more than 2 million people in India and Bangladesh.
The storm hit the Bay of Bengal at midnight local time on November 9, near Sagar Island in Indian West Bengal.
According to local reports, two people have already been killed by the cyclone.
Services at many seaports and airports in the region were suspended – including at the busy Kolkata airport.
Bangladesh’s two biggest ports, Mongla and Chittagong, were closed and flights into Chittagong airport were stopped.
Shah Kamal, Bangladesh’s disaster management secretary, told AFP that the evacuated residents had been moved to more than 5,500 cyclone shelters.
Mamata Banerjee, Chief Minister for the Indian state of West Bengal, tweeted before the cyclone made landfall urging people to stay calm.
She wrote: “Please do not panic.
“Kindly remain calm and co-operate with the administration in its rescue and relief efforts. Be alert, take care and stay safe.”
Forecasters expect Cyclone Bulbul to move north and weaken gradually.
According to the Bangladesh Meteorological Department, the cyclone is set to reach wind speeds of up to 75mph, with gusts of 94mph, and create tidal surges in the sea and rivers when it hits the coastal regions.
Along Bulbul’s predicted path is the Sundarbans, the world’s largest mangrove forest and home to the endangered Bengal tigers.
Indian authorities said military ships and planes have been put on standby to assist with emergencies.
Bangladesh’s low-lying coast is often hit by deadly cyclones, but it has successfully reduced the number of casualties in recent years.