Three sailors have been wounded after the Ukrainian navy said two artillery boats were hit by the strikes in the Black Sea.
The three injured sailors are receiving medical treatment and their lives are not in danger, the FSB said.
Ukraine's ambassador to the UK said Russian special forces had captured two armoured artillery boats and a tugboat in an "act of aggression".
"Today's dangerous events in the Azov Sea testify that a new front of Russian aggression is open," Ukrainian foreign ministry spokeswoman Mariana Betsa said.
"Ukraine is calling now for an emergency meeting of the United Nations Security Council."
The FSB claimed it had "irrefutable evidence that Kiev prepared and orchestrated provocations... in the Black Sea".
"These materials will soon be made public," it added.
Earlier on Sunday, Ukraine accused a Russian coastguard vessel, named the Don, of ramming one of its tugboats, damaging its engine, hull and side railing.
Ukraine's minister of internal affairs posted footage on Twitter purportedly showing the incident.
It allegedly took place as three Ukrainian navy boats - including two small warships - headed for the port of Mariupol in the Sea of Azov, an area of heightened tensions between the countries.
Russia - which claims the waters off Crimea after annexing the peninsula in 2014 - accused Ukraine of illegally entering the area and deliberately provoking a conflict.
It placed a huge cargo ship beneath the Russian-controlled Kerch Strait Bridge to block Ukrainian boats from access to the sea.
"Their goal is clear - to create a conflict situation in the region," the FSB said in a statement.