Investigative Committee of Russia on Sunday, Aug. 21, 2022, investigators work on the site of the explosion of a car driven by Daria Dugina outside Moscow. AP
Dugin -- an outspoken ultranationalist intellectual and a vocal supporter of the Kremlin's offensive in Ukraine -- is thought to have been the likely target of the attack.
"The crime was prepared and committed by Ukrainian special services," the FSB said in a statement carried by Russian news agencies.
It added that the perpetrator -- a female Ukrainian citizen born in 1979 -- on Sunday had fled to EU member Estonia.
The FSB in its statement identified the woman as Natalia Vovk.
Daria Dugina was killed Saturday when a bomb placed in a Toyota Land Cruiser went off as she drove on a highway some 40 kilometres (25 miles) outside Moscow.
Russian President Vladimir Putin on Monday expressed his "sincere condolences and words of support" following Dugina's death.
"A vile, cruel crime ended the life of Daria Dugina, a bright, talented person with a real Russian heart -- kind, loving, sympathetic and open," Putin said in a message to Dugina's family released by the Kremlin.
He added that she "proved with her deeds what it means to be a patriot of Russia".
According to the FSB statement, the attacker arrived in Russia in July 2022 with her underage daughter and rented an apartment in the same building where Dugina lived.
The supposed attacker followed Dugina in a Mini Cooper with registration plates issued in Kazakhstan, Ukraine and in the breakaway Donetsk People's Republic in eastern Ukraine, the FSB added.
The FSB said the attacker was at a festival outside Moscow that Dugin and his daughter had attended on Saturday.
Reports from Russian media suggested Dugina had borrowed her father's car at the last minute.
Dugin, 60, sometimes called "Putin's Rasputin" or "Putin's brain," is an outspoken Russian ultranationalist intellectual.
He has long advocated the unification of Russian-speaking territories in a vast new Russian empire and wholeheartedly supported Moscow's operation in Ukraine.
He was put on a Western sanctions list after Russia annexed Crimea in 2014, a move he also backed.
A Ukrainian presidential adviser, Mykhaylo Podolyak, on Sunday denied that the Kyiv authorities were behind the bombing.