Russia annexed the Black Sea peninsula of Crimea from Ukraine in 2014, drawing sanctions and condemnation from the West. Kyiv wants the territory back.
Vladimir Kulishov, first deputy head of the FSB security service, said Russia had told the vessel not to enter its waters, but that it had crossed what he said was the border on Oct. 13 near southern Crimea.
He said HMS Dragon had invoked the right of "innocent passage," a concept which allows ships to transit the territorial waters of foreign coastal states in a manner not prejudicial to their peace, good order, or security.
Kulishov told Russia's RIA new agency in an interview that Moscow had demanded the vessel immediately leave its waters, to which the warship's captain replied there was poor signal reception.
"The warship was driven out into neutral waters by the joint action of the Russian navy and air force," Kulishov was quoted as saying.
The incident comes as Russia's relations with the West languish at post-Cold War lows.
There was no immediate comment from the British defence ministry.