Ukraine's President Volodymyr Zelenskiy said the strikes had caused significant damage to the oil terminal port. They also hit an oil tanker, a small Russian Karakurt-class missile ship and a patrol boat in the Baltic Sea, he said on Telegram.
"Each such result further limits Russia's war potential," he wrote.
Alexander Drozdenko, governor of the northwest region which hosts the port, said more than 60 drones were downed overnight. He said the fire at Primorsk, a major oil-exporting outlet, was quickly extinguished and there had been no oil spill following the attack.
Among numerous reports of other attacks elsewhere in Russia, the governor of the Belgorod region bordering Ukraine, Vyacheslav Gladkov, said a drone had hit a car, killing a 21-year-old man and his father on the spot.
Primorsk, one of Russia's largest export gateways, has capacity to handle one million barrels per day of oil supply. It has been hit multiple times in recent months as as US-brokered talks to end the Ukraine war have stalled.
Zelenskiy earlier on Sunday said Ukrainian forces also struck two shadow-fleet tankers in waters at the entrance to the Russian Black Sea port of Novorossiysk.
"These tankers had been actively used to transport oil - not anymore," Zelenskiy said on Telegram.
"Ukraine's long-range capabilities will continue to be developed comprehensively - at sea, in the air, and on land."
Kremlin spokesman Dmitry Peskov said global oil prices may rise further if Ukraine continues to hit Russia's oil infrastructure, Russian TV reported.
"If additional volumes of our oil are dropped from the market, prices will rise further from current levels, which are already above $US120 ($A167) a barrel," Peskov said.
"That would mean that even with lower export volumes, our companies would earn more money and the state would receive more revenue."
Moscow regional governor Andrei Vorobyov said on Saturday evening that a 77-year-old man had died in a village following a drone strike. And Sergei Sobyanin, mayor of the city of Moscow, said four drones were downed on their way to the Russian capital.
Vasily Anokhin, governor of the western Smolensk region, said three people, including a child, were injured on Sunday after a drone attacked an apartment block there.
Russian troops were meanwhile inching towards the city of Kostiantynivka in Ukraine's eastern Donetsk region, Ukraine's top army official said on Saturday.
