The plant's owner said the flooding occurred due to a pressure surge in water pipes.
Eight workers were confirmed dead and 14 injured. Investigators initially said 54 people were missing, but local emergency workers said the number could be as high as 65.
The accident shut down the power station, which supplies several major aluminum plants. The government said electricity supplies from other power plants were being re-routed to help cover the shortfall, but outages were reported throughout the region.
The plant's dam, a towering structure that stretches about a kilometer (more than half a mile) across the Yenisei River, was not damaged and towns downstream were in no danger, Emergency Situations Minister Sergei Shoigu said.
The accident produced an oil spill, however, and the slick was floating downriver. The slick had grown to cover as much as 10 square miles (25 square kilometers), the Natural Resources Ministry said.
Two of the plant's 10 turbines were destroyed and a third was seriously damaged, said Vasily Zubakin, acting chief executive of the plant's owner, RusHydro. He said the company was still assessing the state of the remaining seven.
Shoigu said the repairs would be difficult. "We're probably talking about years rather than months to restore three of the 10 turbines," he said on state-run television.
The world's largest aluminum producer, Rusal, was operating as usual, with its smelters being powered from other plants, company spokeswoman Yelena Shuliveistrova said.
The company was in talks with the government about possibly reducing output to free up energy supplies needed elsewhere in the region, Rusal said in a statement.
Half of the residential buildings in Abakan, the capital of the Khakassia region where the plant is located, were left without power. Residents were stocking up on basic supplies and gasoline, Mayor Nikolai Bulakin said on Ekho Moskvy radio.
Abakan, home to 160,000 people, is located about 110 miles (180 kilometers) north of the plant. Power shortages also were reported in the Tomsk and Kemerovo regions.
It was not immediately clear how many people were potentially affected by the accident.
The Sayano-Shushinskaya plant was working at record capacity in June and July due to high water levels in the river, RusHydro reported last week. The plant went into operation in 1978.
Aging infrastructure has long been regarded as a key obstacle to Russia's development.
Analysts have warned that Russia needs to boost its power production significantly to meet the growing demand of industrial producers or it would face regular power shortages as soon as several years from now. Monday's accident put these plans in jeopardy.
RusHydro's stocks were down more than 7 percent on the MICEX on Monday morning when the stock exchange suspended trading for its shares.