Yandarbiyev's teenage son was critically wounded in the blast, which occurred as he and his father were driving away from a mosque where they had performed Friday prayers, said the Interior Ministry and a local hospital.
"We are collecting evidence in order to reach the perpetrators," Qatar's chief of security, Mubarak al-Nasr, said on the pan-Arab satellite channel Al-Jazeera, which is based in the country.
Yandarbiyev, who was acting president of Chechnya in 1996-97, had been linked to the al Qaeda terror group. Russia had been seeking his extradition from Qatar, where he lived for at least three years, accusing him of ties to kidnappers and international terrorists.
Al-Jazeera and fellow Arabic satellite channel Al-Arabiya reported that two people were killed in the explosion. But the Interior Ministry did not confirm this.
An Interior Ministry official said the explosion at 12:45 p.m. killed Yandarbiyev and injured his 13-year-old son, the official Qatar News Agency reported.
A doctor at Hamad General Hospital told The Associated Press that Yandarbiyev died on his way to the hospital. The doctor said his son was in critical condition.
The doctor, who spoke on condition of anonymity, said the father and son were the only casualties brought to the hospital.
There was no immediate claim of responsibility for the blast. Such explosions are almost unheard of in Qatar, a quiet state with tight security.
Last year, the United Nations put Yandarbiyev on a list of people with alleged links to the al Qaeda terrorist group, which is blamed for the Sept. 11, 2001, attacks on the United States. The U.S. government also put Yandarbiyev on a list of international terrorists who are subject to financial sanctions.
Al-Jazeera said the explosion occurred after Yandarbiyev had prayed at a mosque in the upscale residential area of al-Dafnah, a northern suburb of Doha. He got into his private car, and the explosion went off at a road intersection 300 yards away.
The station showed a badly mangled and burned SUV, with only its white fender still recognizable. Security forces and a sniffer dog worked the area as a body, wrapped in white sheet, was loaded into a waiting ambulance.
Born in 1952, Yandarbiyev became vice president of the Russian republic of Chechnya under separatist president Dzhokhar Dudayev, and served as acting president of de facto independent Chechnya in 1996-97. Yandarbiyev headed the rebel delegation to talks with then-Russian President Boris Yeltsin and then Prime Minister Viktor Chernomyrdin in 1996.
Yandarbiyev opened a Chechen embassy in Kabul and a consulate in Kandahar during the reign of the hard-line Taliban militia.
Yandarbiyev denied links between Osama bin Laden's al Qaeda network and Chechen rebels.
Russian forces withdrew from Chechnya in 1996 after a disastrous 20-month war with rebels, leaving the republic largely lawless and running its own affairs. Troops swept in again in 1999 after Chechnya-based militants launched raids into a neighboring region and after some 300 people were killed in apartment building explosions that Russian officials blamed on Chechen separatists.
Yandarbiyev was a nationalist poet and children's book author, and he became one of the most prominent proponents of radical Islam among the rebels. He came in third in de facto independent Chechnya's 1997 presidential elections, behind moderate Aslan Maskhadov and fiery rebel Shamil Basayev.
Yandarbiyev is the most prominent Chechen separatist to have been killed since the 2002 death of warlord Omar Ibn al Khattab, who reportedly was poisoned.
Human rights organizations have criticized Russian tactics in the conflict in Chechnya.
"Russian forces round up thousands of men in raids, loot homes, physically abuse villagers, and frequently commit extrajudicial executions," Human Rights Watch charges. "Those detained face beatings and other forms of torture, aimed at coercing confessions or information about Chechen forces."