At least two people are known to have died and six are missing but the Emergencies Ministry in Moscow has denied initial reports that scores of people were swept out to sea.
The head of the ministry's press service, Marina Rykhlina, confirmed the deaths but said no holidaymakers were hurt when a wall of water swept through camp sites.
The authorities have evacuated 440 people from around the port, whose hinterland is a popular destination for Russian tourists.
Roads have been destroyed in the region and all rail links into the city have been diverted inland to Krasnodar.
Heavy rain had caused the mountain lake of Abrau Durso, just above Novorossiysk, to burst its banks, sending floodwater cascading down the slopes.
The two people killed were found in a flooded village at the lake, which is home to Russia's most famous wine-producer. Another eight people were rescued.
The Emergencies Ministry said it was sending a boat to the coastal village of Paraskeyevka, to check reports of possible casualties there.
Warnings
Storm warnings are in force along the whole of the Black Sea coast of Russia, and local authorities are asking residents and tourists to avoid the sea shore and river banks.
Southern Russia has been hit hard by flooding in recent weeks.
Heavy rains in July left more than 100 people dead, and forced hundreds of thousands more to evacuate their homes.