Cyclone Fani hit the eastern Odisha state, before moving northeastwards into Bangladesh where more than a million people have been moved to safety.
The cyclone lost some of its power after hitting land and was downgraded to a deep depression by the Indian Meteorological Department.
But dozens of villages on Bangladesh’s coast were submerged after a storm surge breached embankments, according to a disaster ministry official in Dhaka.
The storm destroyed several houses in the Noakhali district, where a two-year-old child was killed and about 30 people were injured, local official Tanmoy Das told Reuters.
The cyclone built power over the northern reaches of the Bay of Bengal before tearing into Odisha.
Indian media reported that at least 12 people had died across the state, with most deaths caused by falling trees.
“Destruction is unimaginable… Puri is devastated,” Odisha’s special relief commissioner Bishnupada Sethi said.
The 1.2 million people who had to flee have gone to 4,000 shelters.
The National Disaster Response Force dispatched 54 rescue and relief teams of doctors, engineers and deep-sea divers to flood-prone areas along the coast and as far afield as the Andaman and Nicobar Islands – a union territory about 840 miles east of mainland India in the Bay of Bengal.
The timing of the cyclone is not typical, according to data from the Meteorological Department.
Most extremely severe cyclones hit India’s east coast in the post-monsoon season, but Fani spent 10 days gathering strength over the sea, delivering a huge blow when it made landfall.