In the claim, posted on a website, Basaev also took responsibility for other recent terrorist attacks in Russia.
The claim said brigades of the group "Riyadus-Salikhin" carried out the Beslan attack and what it called other recent "successful military operations."
These include the near-simultaneous bombings of two planes and a suicide bombing outside the Moscow subway last month -- attacks that killed an additional 100 people.
"If the Russian leaders really want me to be in charge of such terrorist acts, they are out of luck and they can't make me do that." -- Aslan Maskhadov
Basaev purportedly expressed regret for the bloody outcome in Beslan, but laid the ultimate blame on President Vladimir Putin.
He said his men had been ready to release their hostages if authorities had met their demands -- an immediate end to the war in Chechnya and a withdrawal of Russian troops, or Putin's resignation.
The claim appeared a day after Russian President Vladimir Putin repeated his position against negotiating with Chechen separatists, and dismissed the Beslan attackers as part of a terrorist "internationale."
"Mr. [Osama] Bin Laden has already offered Europe twice a so-called truce in exchange for the withdrawal of troops from Iraq. Nobody even thinks of entering into negotiations with him. Why? Because the whole civilized world believes that the methods bin Laden and his organization use are such that they do not permit people of today's civilized world to discuss any issues with him whatsoever. So there do exist people with whom one cannot engage in negotiations," Putin said.
The statement also appeared a day after former Chechen President and separatist leader Aslan Maskhadov again denied being behind the hostage-taking. In an interview with RFE/RL, Maskhadov said: "If the Russian leaders really want me to be in charge of such terrorist acts, they are out of luck and they can't make me do that. This is not something I am able to do."
Today's claim of responsibility from Basaev could not be verified.
Earlier, a little-known group called the Islambouli Brigades had claimed the plane and Moscow subway attacks. That group also said it had carried out the attacks in the name of Chechen independence.
Basaev is held responsible for the 2002 "Nord-Ost" theater hostage-taking in Moscow. Some 170 people, including the 40 militants, died when Russian forces stormed the theater.
If Basaev's claim this time is authentic, it provides more details of who was behind the Beslan tragedy.
The statement says 33 rebels were in the group. This included Chechens, Ingush, Ossetians, and Russians, as well as two Arabs.
It also says the group has no connection to Al-Qaeda.
Irina Isakova is a Russia expert and former adviser to both the Russian and British parliaments.
She declined to comment directly on Basaev's claim, but said any evidence of foreign involvement in the attacks -- however small -- would boost Russian authorities' argument that the Beslan attackers are part of an international network.
"[The numbers are not essential] but [any Arab connection would be important] at least for those who are trying to curb the violence there. So I think that even if there is a slight connection [with Arabs or Al-Qaeda] that gives a very strong argument to those saying that it is part of the wider international picture," Isakova said.
Another detail in the statement is the relative low cost of the attacks -- it says the Beslan attack cost 8,000 euros ($9,769) and the airplane bombings cost $4,000.
Colonel Nick Pratt heads the Program on Terrorism and Security at the George C. Marshall European Center for Security Studies in Germany.
He says international funding for terrorist groups in general has dried up in the last few years. "Money is drying up, but it doesn't cost a lot of money to run these operations. The cost of the operation that Basaev boasted about would be very inexpensive, you have to bribe some people to get through checkpoints, you need a couple of vehicles, they already had the weapons and explosives," Pratt said.
Other details of the siege are also emerging.
"The Guardian" today reported that Putin had been prepared to make a deal with the hostage-takers to save the lives of the children held hostage.
The paper quotes Aslanbek Aslakhanov, the president's adviser on Chechnya, as saying he arrived in the region on 3 September ready to negotiate. The deal would reportedly have involved the release of militants held in Ingushetia and an offer to swap up to 700 adults for the children held hostage.
But he said he was too late -- shooting and explosions had already broken out at the school.