A storm on Sunday wrecked two vessels, with one sinking after being split in half and another severely damaged boat running aground.
One crew member died from hypothermia and 26 others were rescued from the damaged tankers, the Russian government said.
"Anapa is in a state of emergency due to the oil products spill," the local mayor's office said on Telegram, adding that "an operation to eliminate the consequences is underway".
Located in the southern Krasnodar region close to the Crimean peninsula annexed from Ukraine, the town of Anapa has a population of around 90,000 people.
The regional minister for emergency situations, Sergei Shtrikov, on Tuesday said fuel dispersed over at least 30 kilometres (19 miles) of beach, contaminating the coastline.
"Oil slicks from the sunken tanker have reached the coastline of the Anapa district," said Svetlana Radionova, head of Russia's natural resources agency.
Authorities also declared a state of emergency in four smaller towns.
The emergency situations ministry said 400 personnel joined clean-up efforts.
Videos posted on the Telegram channel Baza showed stretches of beach and birds covered in thick black petroleum.
Ukraine's environmental ministry said on Monday that the "significant" accident posed the "threat of a large-scale environmental disaster in the Black Sea" and could have "catastrophic consequences for the marine ecosystem".
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