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The fire broke out as welders were working during maintenance in the first power unit. A thick plume of black smoke was seen from the upper deck.
The area affected by smoke and fire covers 120 square meters (1,292 square feet), TASS said.
The warship made its first combat deployment in 2016, in the Mediterranean Sea, when its fighter jets delivered strikes in Syria.
But the carrier has had problems since its return to Russia for maintenance.
The accident took place after a power supply disruption caused the pumps on the floating dry dock, which was holding the aircraft carrier, to break down, causing the structure to sink.
The dock, known as PD-50, was one of the world's largest floating dry docks and the biggest in Russia, measuring 330 meters long (1,082 feet) and weighing more than 80,000 tons.
Following the disruption, PD-50 quickly became submerged, causing two cranes to fall onto the deck of the Russian vessel, leaving a hole measuring 4 meters by 5 meters (13 feet by 16.4 feet), state-run media Sputnik said.
After that accident, Russian officials told state media the damage would not delay repairs on the Admiral Kuznetsov, which were due to be completed by 2021.
The Kuznetsov accident is at least the second deadly fire for Russia's navy this year.