However, the accident has created a large oil slick that is now floating down the Yenisei river, which flows north through Siberia to the Arctic.
Officials said towns downstream of the plant were not thought to be at risk.
Russian President Dmitry Medvedev has ordered Emergencies Minister Sergei Shoigu and Energy Minister Sergei Shmatko to fly to the scene and take personal control of the crisis.
RusHydro, the operator of the power station, said the damage would run into "billions of roubles" and would take several months to repair.
In a statement, the Kremlin said the accident had been due to a "hydraulic impact" at the plant on Monday morning, which had forced the shutdown of all 10 of its power units.
An investigation is under way to determine the exact cause.
Andrei Klyuvev, an official at the emergency situations ministry, said rescue divers had pulled out one person from a flooded room underneath the turbine hall, but said many more could be trapped.
Towns downstream of the station's dam were said not to be at risk
"At the moment we cannot determine whether these people were down there or managed to get out somewhere but we know that there were that many people on this shift," he told Ekho Moskvy radio.
Water was currently being pumped out of the turbine room, and there was "no danger of the accident developing further", a local official co-ordinating the rescue operation, Roman Strelnikov, told the Associated Press.
The natural resources ministry said it was concerned by the environmental impact of the accident, which had caused a 5km (3 mile) oil slick along the Yenisei river.
"According to preliminary data, transformer fluid has leaked from one of the hydroelectric station's damaged units," the ministry said.
Mr Shoigu said repairs would be difficult and take some time.
"We're probably talking about years rather than months to restore three of the 10 turbines," he told state television.
Major aluminium plants nearby were forced to switch to alternative sources of electricity after the accident.
It also prompted shares of the plant's owner, RusHydro, to drop by more than 15% on the London Stock Exchange, while trading was suspended in Moscow.
The Sayano-Shushenskaya power station is located in the Siberian region of Khakassia, some 3,000 km (1,875 miles) east of Moscow.
The dam above it is 245m (800ft) high and stretches 1km (0.6 miles) across the Yenisei river.
Opened in 1978, the station provides a quarter of RusHydro output and is a major supplier of power to at least two smelters owned by United Company RUSAL, the world's largest aluminium producer.
UC RUSAL said all its plants were operating as normal with alternative power supplies.
"I think that owing to the new energy supply scheme, output will be ensured. Nevertheless, the government must keep this issue under control," President Medvedev said in a statement.
"I hope that all the managers dealing with this issue as well as the leadership of the government will make the necessary effort, and take all the necessary decisions to implement these plans," he added.
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