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.