A large group of sailors is seen on parade in the Crimean port city of Sevastopol, met by Navy Commander-in-Chief Adm Nikolay Yevmenov.
The Moskva was the flagship of the Black Sea Fleet.
The ship's demise has been described as a huge blow to Russian morale.
The ministry announced late on Thursday that the Moskva had sunk on stormy seas after a fire caused by exploding ammunition. Ukraine said it had sunk the ship with two Neptune missiles.
Russia said at the time that the crew had been taken to Sevastopol, a major port Crimea, which Russia annexed from Ukraine in 2014. This, however, is the first time any evidence has been offered of the survival of any sailors.
The video shows Adm Yevmenov and two other officers standing on a parade ground in front of about 100 sailors.
It is not clear when the meeting took place.
In an interview shown at the end of the video, Adm Yevmenov says that the officers and crew are currently residing at their base in Sevastopol and will continue their service in the navy.
Russia has not reported any casualties from the fire or the sinking.
However, an unnamed US official quoted by Reuters said Washington believed there were casualties.
Ukraine says the ship's captain, Anton Kuprin, was killed on board, but the BBC has been unable to verify the claim.
The 12,490-tonne cruiser is the biggest Russian warship to be sunk in action since World War Two.
On the first day of Russia's invasion of Ukraine on 24 February, the Moskva gained notoriety after calling on a small garrison of Ukrainian border troops defending Snake Island in the Black Sea to surrender - to which they memorably radioed an expletive-laden message of refusal.
Originally built in the Soviet era, the Moskva entered service in the early 1980s. The vessel was actually built in Ukraine's southern city of Mykolaiv, which has been heavily bombed by Russia in recent days.
The guided missile cruiser was previously deployed by Moscow in the Syria conflict where it supplied Russian forces in the country with naval protection from the Mediterranean.
It is the second major vessel Russia has lost since the start of its invasion of Ukraine.