Cyclone Fani, one of the most severe storms to hit the region in recent years, made landfall at 08:00 local time (02:30 GMT) on Friday.
More than one million people have been evacuated from the eastern state of Orissa, also called Odisha.
A state official said two people had been killed.
Flooding has also been reported in several areas, and forecasters say a storm surge of 1.5m (5ft) could threaten low-lying homes.