Local media reports quoted state government officials as saying that while one person died when a tree fell on him, another died on heart attack inside a cyclone shelter, and a woman died when flying debris from a concrete structure hit her.
A large number of trees and electricity poles were uprooted, and power supply was cut off in many parts of the state. Many areas in Puri of Odisha and other places were submerged as heavy rain battered the coast.
Strong winds were seen in parts of the southern state Andhra Pradesh and eastern state West Bengal.
The cyclonic storm was said to be weakening gradually.
The Odisha government has evacuated over 1 million people from the coastal areas to camps on higher grounds. The Indian army, navy, air force, coast guard and disaster management agencies are on stand-by.
Prime Minister Narendra Modi said over 1,000 crore Indian Rupees (around 143 million U.S. dollars) had been released in advance for the states affected by the cyclone.