She said two people had been killed, including one dragged into the sea by the waves in Digha and another crushed when his house collapsed.
The storm packed lashing rain and winds gusting up to 155 kilometres (96 miles) an hour, the equivalent to a category two hurricane.
“I have never seen such a storm ever in my life,” said Digha resident Purnendu Jana. “The water may cross the main road for the first time.”
Local hotel owner Shiuli Das said: “Many of us are here, all of us are really scared.”
Nearly 20,000 houses were damaged and more than a dozen river islands were flooded, with a number of embankments breached, Banerjee said.
In Odisha, a young man and a priest were killed in separate incidents after they were crushed by falling trees.
There was extensive damage with hundreds of trees uprooted, some bringing down power lines, relief official Pradeep Kumar Jena said.
Some thatched homes were also damaged during the storm, but telecommunication networks were not affected, he added.
The state’s chief minister Naveen Patnaik said aid would be provided to more than 100 villages cut off by the tidal surge.
In neighbouring Bangladesh, one man was killed by a falling tree as the waves smashed through water defences and inundated thousands of homes, officials told AFP.
The Bangladesh military added that 12 crew on board a cargo ship carrying stones in the Bay of Bengal were rescued after it sank amid the extreme conditions.
Locals feared the situation would worsen in the evening as the tide rises to a higher level than normal because of a full moon.
“My house is already submerged under four feet of water. I don’t know what will happen during the full moon tonight,” Mostak Ahmed told AFP by phone from Khulna, in the country’s south.
Almost 5,000 disaster workers were deployed in India with tree and wire cutters, emergency communications, inflatable boats and medical aid, the National Disaster Response Force said.
Officials fear the storm will further complicate efforts to halt a surge in coronavirus cases that has now killed 310,000 people in the country.
Masks were distributed in emergency shelters, but West Bengal state minister Bankim Chandra Hazra told AFP that maintaining social distancing would be “a big challenge”.
“This cyclone spells double trouble for millions of people in India as there is no respite from Covid-19,” said Udaya Regmi from the International Federation of Red Cross and Red Crescent Societies.
Some vaccination centres in threatened districts, as well as Kolkata, suspended operations and a special effort had been launched to ensure the supply of oxygen and medicines to hospitals, officials said.
Some of the deadliest storms in history have formed in the Bay of Bengal, including one in 1970 that killed half a million people in what is now Bangladesh.
Odisha’s worst-ever cyclone, in 1999, killed 10,000 people.
Last year, Cyclone Amphan, the worst since then, caused widespread devastation but timely evacuations meant there were fewer than 150 fatalities.