According to the pro-Chechen website Kavkazcenter.com, the former Chechen prime minister also warned further attacks will follow.
"Our martyrs' fighting brigade carried out two attacks under our operation Boomerang in Yessentuki (Ingushetia) and in Moscow, aimed at forcing the Russians to make peace," Basayev said.
He added: "I emphasize that we did not aim to terrorise anyone. Our goal was to wipe out the accomplices to genocide of the people of Chechnya."
'Russia torture centre'
The Pyatigorsk region of southwest Russia, where a bombing killed 46 people on 5 December, was chosen "because that is the site of the Belaya Lebed detention centre where hundreds of Chechen hostages are held", he said.
"They are victims of inhuman torture... several of them have disappeared without trace."
Turning to the 9 December attack in Moscow, Basayev said: "The target of our martyred sister in Moscow was the Duma," the lower house of parliament for which elections were held two days earlier.
Vladimir Putin launched the
second Chechen war in 1999
It was possible, he said, that "something prevented her from walking the 50 metres to her goal" before setting off the explosives.
The attack killed six people and wounded 14 others following legislative elections in which President Vladimir Putin's allies won a sweeping victory.
Basayev, 38, is Russia's most wanted man.
He rose to prominence when Russian forces invaded Chechnya in 1994, and soon became one of the leading commanders of the Chechen resistance.
At the end of the war he stood for the Chechen presidency but came second to Aslan Maskhadov.
He was then appointed Chechen prime minister, but his popularity waned as crime and kidnappings spiralled in the internationally isolated republic.