Cyclone Yaas was packing gusts of up to 140 kph (87 mph) as it hit land, authorities said, days after Tauktae tore up the western coast, triggering mass evacuations and piling pressure on authorities battling a deadly second wave of the coronavirus.
Authorities said more than a million people had been moved out of the storm's path, while television broadcast images of rough seas, strong winds and rains lashing the state of Odisha, with shops and homes boarded up.
By noon, the "very severe cyclone" would cross Odisha and its neighbor, West Bengal, weather officials said.
West Bengal's chief minister, Mamata Banerjee, told reporters that about 20,000 mud houses and temporary shelters had been damaged in the state.
"I have not seen anything like this before," said another state minister, Bankim Hazra, after seawater gushed into the low-lying areas of Sagar island in the Bay of Bengal and the tourist town of Digha, where a police station was flooded.
"Successive high tides battered the coastline," he added. "It is inundation all around and villages are cut off."
The state's Kolkata airport was closed to flights until Wednesday evening.
Cyclones in the Bay of Bengal are common at this time of year, and often roar ashore, bringing death and destruction to the coastal areas of both India and neighboring Bangladesh.
Police said they had rescued 10 people after their boat capsized near shore in Odisha's Jagatsinghpur district on Tuesday. Naval base Chilka is monitoring ships in the area and is ready for rescue operations, the Indian Navy said.
The devastating wave of virus infections complicated storm preparations. Odisha officials said they had suspended testing, vaccination and a door-to-door health survey in the three districts in the storm's path.