Russia's President Vladimir Putin condemned what he described as a "stupid criminal action" by the Ukrainian authorities and vowed to take additional steps to ensure security of Crimea.
He also strongly urged the West to warn Kiev against "resorting to terror instead of searching for a peaceful settlement."
Ukrainian President Petro Poroshenko rejected the Russian claims as "fantasy" and "provocation," saying in a statement that his government would use only political and diplomatic means to restore its sovereignty over Crimea.
The Federal Security Service, known under its Russian acronym FSB, said in yesterday's statement that its officer was killed over the weekend near Armyansk within a few kilometres (miles) of the de-facto border between Crimea and Ukraine when FSB officers engaged in a gun battle with a group of "saboteurs" from Ukraine.
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The FSB said the intruders carried an arsenal of bombs, ammunition and mines.
The FSB said it also busted what it called a network of agents of Ukrainian military intelligence in Crimea, and detained several people, including a Ukrainian citizen identified as Yevgeniy Panov, whom it described as a Ukrainian military intelligence officer.
The agency claimed that the Ukrainian intelligence operation had sought to destabilise the situation in Crimea ahead of Russia's parliamentary elections set for next month.
Putin, speaking at a news conference in Moscow, accused the Ukrainian leadership of engaging in "terror" instead of discussing peace settlement in eastern Ukraine.
Poroshenko countered in his statement that "we would never ever use terror to de-occupy Crimea."
The Ukrainian government said over the weekend that Russia briefly closed its border crossings with mainland Ukraine, and social media users earlier this week posted photos and videos of dozens of armoured vehicles on Crimean highways heading toward the de-facto border.