Dozens of patients were taken to other medical facilities in the city.
About 120 firefighters were mobilised to tackle the blaze, which was later brought under control. PM Narendra Modi said he was "anguished" by the fire.
Eyewitnesses and hospital sources said the fire, suspected to have been caused by an electrical short circuit, was first spotted in the ICU of the medicine ward before spreading to the dialysis ward next door.
Most of the victims reportedly died from smoke inhalation.
"The fire left 20 people dead and more than 100 people injured. They have been admitted to various hospitals in Bhubaneswar and the medical college in Cuttack," the state`s health secretary, Arati Ahuja, told the BBC`s Sandeep Sahu.
Many of the injured are in a critical condition. Earlier reports had put the death toll at 23.
"Deeply anguished by the loss of lives in the hospital fire in Odisha. The tragedy is mind-numbing. My thoughts are with bereaved families," Prime Minister Narendra Modi tweeted.
In 2011, 89 people died in a fire that broke out in a hospital in the eastern city of Kolkata.
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