The civilian port, which handles two per cent of the world's oil supply and also exports grain, temporarily halted all ship movement before resuming normal operations, according to the Caspian Pipeline Consortium which operates an oil terminal there.
Russia's Defence Ministry said in a short statement the Ukrainian attack by two sea drones had been repelled in the waters outside the base and the drones had been destroyed.
It made no mention of any damage.
But a Ukrainian intelligence source said the Olenegorsky Gornyak, a Russian Navy landing ship, had suffered a serious breach and could not conduct combat missions following what they said was a joint operation by the Ukrainian Navy and SBU intelligence service.
Separately, a source with knowledge of the port's operations said a large Russian naval vessel had to be towed ashore because it could not move under its own power after being damaged.
The source, who did not name the vessel, said oil and grain loadings were still taking place at the port which resumed normal operations hours after the attack.
Reuters was not able to immediately and independently verify the assertions about damage to the Olenegorsky Gornyak warship.
There was no immediate comment from Moscow.
Unverified video footage published on social media purported to show damage to it.
In one video, it was seen listing heavily to its port side.
Andrei Kravchenko, a Novorossiysk city official, said on the Telegram app the Olenegorsky Gornyak was one of two ships that had "instantly reacted" to fend off the Ukrainian attack.
Russian social media users had reported hearing explosions and gunfire near Novorossiysk on Friday morning.
"Tonight, the armed forces of Ukraine attempted to attack the Novorossiysk naval base with two unmanned sea boats," the Defence Ministry said in its statement.
"In the course of repelling the attack, the unmanned boats were visually detected and destroyed by fire from the standard weapons of the Russian ships guarding the outer waters of the naval base."
The Caspian Pipeline Consortium which loads oil onto tankers in Novorossiysk said its facilities had not been damaged and oil loadings continued onto tankers which were already moored.
The port is one of the biggest in the Black Sea, and a major terminal for the export of oil and oil products. The Caspian Pipeline Consortium is the main exporter of Kazakh crude oil.
Exports of Russian and Kazakh oil from Novorossiysk average about 1.8 million barrels per day.
Britain's military intelligence service said last September Russia's Black Sea Fleet had relocated some submarines from the Crimean port of Sevastopol to Novorossiysk, probably because of the increased threat of Ukrainian long-range strikes.
Tensions in and near the Black Sea have escalated since Russia last month withdrew from a deal allowing the safe export of grain from Ukrainian ports.
Russian drones and missiles have since repeatedly struck Ukrainian port facilities and grain silos on or near the Black Sea.
Russia has also reported an uptick in attempted Ukrainian naval drone attacks on its own warships in the Black Sea.
The Russian Defence Ministry said on Friday its forces had also repelled an attempted Ukrainian attack on Crimea, which it annexed from Ukraine in 2014, using at least 13 aeroplane-like drones.
It said there were no casualties or damage.
