Around 130 others were hurt in the fire, which comes just weeks before thousands of athletes and foreign visitors are expected in the country for the Winter Olympics.
While South Korea has risen to become the world's 11th- largest economy, some of its infrastructure was built rapidly and it has a history of preventable disasters.
It was the country's second major blaze in a month, and officials admitted there was no sprinkler system installed at the hospital.
The death toll rose rapidly throughout the morning as some of those initially pulled from the blaze succumbed to their injuries. At one point the presidential Blue House put it at 41, before authorities lowered it to 37, blaming double-counting.
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Three of the dead were medical staff, officials said.
"Two nurses said they had seen fire suddenly erupting in the emergency room," said fire chief Choi Man-Woo.
All the dead were in the hospital, Choi said.
Video footage and pictures showed the building engulfed by thick, dark smoke and surrounded by multiple fire trucks.
Survivors were brought out wrapped in blankets, and firefighters picked their way through the blackened shell of the building after the blaze was extinguished.
Jang Yeong-Jae, a surviving patient, said he was on the second floor when nurses screamed "Fire!" in the hallway and urged people to leave through the emergency exits.
"Everybody was running around in panic, falling over and screaming as smoke filled the rooms," he was quoted as saying. Jang tore open window screens and escaped on a ladder erected by firefighters.
"There were so many aged patients on other floors... I wonder if they escaped safely," JoongAng quoted Jang's wife as saying.
Hospital director Son Gyeong-Cheol admitted there was no sprinkler system in the building.
None had been required under fire prevention laws, he told journalists, but the hospital had been planning to install them in the coming week to comply with new regulations coming into force in June.