Amphan made landfall on Wednesday, lashing coastal areas with ferocious wind and rain. The storm is weakening as it moves north into Bhutan.
Thousands of trees were uprooted in the gales, electricity and telephone lines brought down and houses flattened.
Many of Kolkata's roads are flooded and its 14 million people without power.
The storm is the first super cyclone to form in the Bay of Bengal since 1999. Though its winds had weakened by the time it struck, it was still classified as a very severe cyclone.
Coronavirus restrictions have been hindering emergency and relief efforts. Covid-19 and social-distancing measures made mass evacuations more difficult, with shelters unable to be used to full capacity.
What's the latest on the cyclone?
Amphan began hitting the Sundarbans, a mangrove area around the India-Bangladesh border home to four million people on Wednesday afternoon, before carving north and north-eastwards towards Kolkata, a historic city that was the capital of the British Raj and widely known as Calcutta.
Parts of West Bengal and Orissa (also known as Odisha) states in India, and areas in south-west Bangladesh, bore the brunt, with winds gusting up to 185km/h (115mph).
At least 72 people have died in India's West Bengal state, and 12 deaths have been confirmed in Bangladesh.
West Bengal chief minister Mamata Banerjee said the devastation in Kolkata, the state capital, was "a bigger disaster than Covid-19".
"Area after area has been ruined," Ms Banerjee was quoted as saying by the Press Trust of India news agency. "I have experienced a war-like situation today."
Three districts in West Bengal - South and North 24 Parganas and East Midnapore - were very badly hit.
In Bangladesh, there are reports of tens of thousands of homes damaged or destroyed and many villages submerged by storm surges in low-lying coastal areas like Khulna and Satkhira. The authorities evacuated nearly 2.5 million people ahead of the cyclone.
Initial assessments of the damage are being hampered by blocked roads and flooding in all these areas.