The cause of the accident was still unclear, but officials said a faulty turbine and increased pressure in the pipes could have triggered the blast.
The claim posted on a website purportedly used by the representatives of Islamist groups said that Riyadus Salikhiin had begun an "economic war" against the Russian government.
"To carry out this task, sabotage groups were set up and sent to numerous regions of Russia with the aim of carrying out acts of industrial sabotoge," the statement said.
"Praise to Allah, on August 17 an act of sabotage was carried out at the Sayano-Shushenskaya hydroelectric plant that had been carefully thought out and prepared for a long time," the statement said.
The group is believed to seek to end Russian rule in the North Caucasus and the establishment of an Islamic emirate in its place.
It is designated as a banned terrorist group by the US state department and in June claimed responsibility for a bomb attack that badly injured Yunus-Bek Yevkurov, the president of Ingushetia.
Analysts were sceptical about the message, which came amid a wave of attacks in the Caucasus region.
"In the case of terrorist attacks, notification about who has carried out an attack usually happens at the time of the attack," Yulia Latynina, a journalist who specialises in the North Caucasus, said.
"The dam accident happened on Monday morning. If this website had placed the announcement half an hour before that, their claim would have been plausible."
Alexei Malashenko, a Caucasus expert at the Moscow Carnegie Centre, described the claim as "propaganda".
He said Chechen rebels had in the past taken responsibility for events generally accepted as accidents, including a fire in Moscow's Ostankino television tower and a mass blackout in the capital.
The internet statement came as Vladimir Putin, the Russian prime minister, visited the scene of the disaster on Friday.
"There's no need to pretend that there's something unclear about this to someone," he said.
Putin also announced that families of those still missing after the accident would be treated the same as families of those confirmed dead.
"Families both of the dead and the missing will receive compensation of one million rubles [$31,000] from the federal budget," he said on television.
The power station has been shut down since the powerful explosion that blew out walls and caused the turbine room at Russia's largest power plant to flood.
Repairs are estimated to take from two to four years, and the energy minister said it would cost $1.2bn to rebuild the turbine room.