The Ukrainian president calls for emergency session of his war cabinet after accusing Russia of capturing three of its ships.
Two sailors have been wounded after a small artillery boat was hit in the strikes, the Ukrainian navy said.
Sky's defence and security correspondent Alistair Bunkall said the Ukrainian president has called an emergency session of his war cabinet in response to the incident.
"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 emergency meeting of United Nations Security Council."
Ukraine accused a Russian coastguard vessel, named the Don, of ramming one of its tugboats in "openly aggressive actions".
The incident 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," Russia's Federal Security Service (FSB) said in a statement.
But the Ukrainian navy said Russia had been informed in advance about the planned journey.
"Russian coastguard vessels… carried out openly aggressive actions against Ukrainian navy ships," its statement said.
A 2003 treaty designates the Kerch Strait and Sea of Azov as shared territorial waters but Russia has been asserting greater control over the area since 2015.
In September, the Ukrainian navy accused Russian border guards of "acts of provocation" against its ships taking the same route.
Ukraine has increased the number of navy ships and border guard patrols in the Sea of Azov, which is reached via the Kerch Strait between Crimea and Russia.
Kiev says its naval build-up is due to Russia stepping up controls in the area this year and carrying out lengthy checks on commercial shipping.