Heavy rains caused flooding across Kolkata, the capital of West Bengal, as winds of up to 100kph brought down trees and communication lines.
"Our village is submerged, we are living in camps and have no clue what further calamity awaits us," Anil Krishna Mistry, a villager, told Reuters by telephone from Bali in the Sundarbans.
In Bangladesh, at least seven people were killed and thousands of homes were damaged by tidal waves triggered by the cyclone, officials said.
Most of the storm surges occurred in the Khulna district near the Sundarbans and the Bay of Bengal.
"Thousands of families have been moved to shelters and many left on their own," Salahuddin Chowdhury, an official of the Cyclone Preparedness Centre in Chittagong, said.
The ports of Chittagong and Mongla were closed as the storm approached.
Cyclones and tropical storms are common in Bangladesh, killing many people in the delta nation of 150 million every year and causing huge damage to crops and property.
Cyclone Sidr swept parts of the Bangladesh coast in November 2007, killing about 3,500 people. A cyclone also killed about 140,000 people in April 1991.