Phailin, the fiercest storm to hit India in more than a decade, was blamed for at least six deaths as it snapped trees and utility lines, blew roofs off homes and flattened crops as it pummeled the region, The Hindustan Times reported.
The newspaper said besides the six confirmed killed, 18 fishermen were missing. Officials had warned all boats not to leave port as the typhoon approached.
A Home Ministry official in New Delhi told the Times an official assessment of the damage would come Sunday morning.
"Till then, we can only keep our fingers crossed," he said.
Heavy rains accompanied the storm as it hit Gopalpur shortly after 9 p.m. local time, the Times reported.
Phailin has been rated a Category 4 storm on a scale of 1-5.
As many as 500,000 left their homes ahead of Phailin making landfall, the BBC reported.
Phailin is a powerful storm that has been compared to Hurricane Katrina, which hit New Orleans in 2005, the BBC said. A similarly powerful storm hit Orissa in 1999, killing more than 10,000 people, but officials say they were better prepared this time.
Phailin was expected to cause a 10-foot storm surge that officials say could cause "extensive damage" to mud houses on the coast.
Anyone in mud and thatched houses in the coastal areas was under mandatory evacuation orders, said Surya Narayan Patra, Orissa's disaster management minister. The army was on standby for emergency operation, and food packages were ready to be dropped into storm-affected areas by helicopter.