Investigators, who described as a journalist and political expert, said they had opened a murder case and would be carrying out forensic examinations to try to determine exactly what had happened.
They said they were considering "all versions" when it came to working out who was responsible for the crime.
Alexander Dugin, Darya's father, has long advocated the unification of Russian-speaking and other territories in a vast new Russian empire.
He wants that empire to include Ukraine where Russian forces are currently carrying out what Moscow calls a "special military operation" to demilitarise Ukraine.
The influence of Dugin, who is on a US sanctions list, over Russian President Vladimir Putin has been a subject for speculation, with some Russia watchers asserting that his sway is significant and others calling it minimal.
Darya Dugina, who also went by the surname Platonova and was reported by Russian state media to be 30 years old, broadly supported her father's ideas and appeared on state TV in her own right to offer support for Russia's actions in Ukraine.
Reuters