A massive operation to evacuate more than a million people to safety from low-lying areas in Orissa state has been put in place.
Next in Fani's sights was West Bengal's capital Kolkata, home to 4.5 million people, with the eye of the storm due around midnight (1830 GMT) and rain already falling hard several hours before.
A state official said two people had been killed.
Storm surge of 4-5 feet height above normal astronomical tide may hit the coastal districts under the influence of the storm and the new moon phase, the Met Office said.
Images shared online showed terrifying winds tearing through deserted streets, destroying street stalls as torrential rain lashed trees.
Odisha special relief commissioner Bishnupada Sethi said no deaths or injuries have been reported.
According to graphics of Joint Typhoon Warning Centre and weather blog Windy's simulation, the storm will enter Bangladesh through Rajshahi-Meherpur region before crossing Rangpur to weaken further into a land depression.
Winds brought by the cyclone were reportedly gusting at up to 205 kilometers (127 miles) per hour.
Reviewing the relief efforts, the cabinet secretary directed that officials of central ministries and agencies should remain in close touch with the Odisha government and provide all required assistance expeditiously. A major hospital in the city suffered extensive structural damage but all patients and staff were safe, authorities said.
"It's a total mess in islands of the Sunderbans as the cyclone has destroyed everything in its path, fueling fears rivers could burst their banks and leave vast areas underwater", said Manturam Pakhira, Sunderbans affairs minister. Meanwhile, the storm also had a string impact in cities like Kharagpur and Burdwan as trees were uprooted and metal hoardings gave way.
It is a "very, very scary feeling", said Tanmay Das, a 40-year-old Bhubaneswar resident, who described "the sound of wind as if it will blow you away".
"From April 25 onwards we (the Indian Meteorological Department and Skymet) had been monitoring the track and intensity of the cyclone continuously, what time it would make landfall and the probable points of landfall", Palawat told AFP.
"Whether the government will seek help in building [houses] will be decided after completing an assessment of the nature and extent of damage", Dash said.
The storm is not expected to touch the country's southeastern district of Cox's Bazar where almost a million Rohingya Muslims are sheltered. This contribution is Hyundai's way of expressing solidarity with the people of Odisha & West Bengal, as they rebuild their lives in the aftermath of the recent devastating Fani Cyclone.