But separatist rebels in Chechnya have insisted Basayev died accidentally when a truck carrying explosives blew up.
The FSB security service chief Nikolai Patrushev said: "Overnight, a special operation was carried out in Ingushetia ... during which Shamil Basayev and other bandits who were preparing an attack were eliminated."
Mr Putin said: "This is a just punishment of the bandits for our children in Beslan, in Budyonnovsk, for all the terrorist attacks that they carried out in Moscow and other regions of Russia including Ingushetia and the republic of Chechnya.”
Television news programs did not broadcast footage of Basayev's body as they have done following the killings in the past 18 months of several other top Chechen rebel figures, but officials indicated this may have been because his body was badly maimed.
Chechen rebels confirmed the death but denied Russian special forces were responsible.
"The Chechen commander died following the accidental explosion of a truck transporting explosives on July 10, 2006, near the village of Ekazhevo, in Ingushetia," Chechnya's separatists said in a statement published on a website, Kavkazcenter.com.
Bashir Aushev, deputy prime minister of Ingushetia, said that Basayev, 41, was among suspected Chechen militants killed as they sat in cars alongside a large truck packed with explosives and detonated in the Ingush village of Yekazhevo.
"Basayev was identified by fragments of his body. To the best of my knowledge, they identified him by his head," Interfax quoted Mr Aushyev as saying.
Ramzan Kadyrov, prime minister of the Kremlin-backed government of Chechnya, said: "The liquidation of Basayev is an unbelievable success.”
"I'm only sorry that I could not take part in his liquidation myself. He was not just the enemy of all of Russian society but he was my personal enemy ... this was a terrorist on an international scale."
Basayev had been the target of one of the most intensive and prolonged manhunts ever mounted in Russia, and a 10 million dollar bounty had been offered for information leading to his capture or death.
Commenting on the death, US President George W. Bush said : "He deserved it."
"If he's, in fact, the person that ordered the killing of children in Beslan, you know, I think he deserved it," Mr Bush said.