"Nineteen bodies were found at the site of the fire, two other people succumbed to their wounds at the hospital," a ministry spokesman said.
The hospital building, made of wood, was destroyed by the fire, the spokesman said.
A further 20 people were hospitalised.
The fire broke out shortly before midnight (local time) at the hospital in the village of Alferovka, which lies in the region of Voronezh in the south of the country.
It took more than 440 firefighters and emergency workers, who arrived at the scene aboard 80 vehicles, around three hours to bring the fire under control.
It was not immediately clear what triggered the blaze, AFP reported.
The fire was the latest tragedy to hit a psychiatric institution in Russia, where outdated Soviet-era infrastructure is still in widespread use and managers often take a lax approach to fire safety.
Scores of people also die in house fires each year.
A fire at a psychiatric hospital in north-west Russia in September 2013 left 37 people dead while another blaze in April of the same year killed 38.
In 2009, 156 people were killed in a nightclub fire in the city of Perm, 1,200 kilometres east of Moscow in one of the deadliest accidents in Russia's modern history.