Winds of more than 125 miles per hour lashed the coastline, where 10,000 people were killed in a cyclone in 1999.
The tornado, which was followed by a powerful hail storm, centred on Kendrapara, around 60 miles from the state capital Bhubaneswar. The official death toll on Tuesday night stood at eight, but unconfirmed reports indicated up to 20.
Rescue workers rushed to the scene as officials said more than 300 homes were destroyed in the storm. Eleven villages were flattened, while electricity and communications were cut.
"Eight bodies have so far been recovered after the tornado hit 11 villages, leaving scores of people injured," said Ranjan Kumar Das, sub-collector of Kendrapara.