The video, filmed a few weeks ago, shows the sailors singing an Indonesian hit called Sampai Jumpa (see you later).
Officials confirmed on Sunday that the KRI Nanggala - which sank off the coast of Bali on Wednesday - had been found split into three pieces on the sea bed.
All 53 crew were confirmed as dead.
The submarine had been taking part in a torpedo drill just before it disappeared and the reason for its sinking is not yet clear.
The video shows the vessel's commander, Heri Oktavian, among those gathered around a crew member playing an acoustic guitar.
"Even though I'm not ready to be missing you, I'm not ready to live without you. I wish all the best for you," they sing.
The recording was made as a farewell for the outgoing commander of the navy's submarine corps, military spokesman Djawara Whimbo told AFP news agency.
Meanwhile, the navy says it is making plans to salvage the vessel and the bodies of the crew.
The wreck's location is more than 800m (2,600ft) deep and experts say recovering it from that depth will require specialist equipment.
"We will analyse the under water pictures and video, the current etc, to decide the technology that will be used," First Admiral Julius Widjojono told reporters.
President Joko Widodo has described the crew as Indonesia's "best patriots" and pledged government funding for the education of their children.
"The country will award them [the crew] by promoting them one rank higher and awarding them with a star medal for the services and contribution of these warriors," he said. "The government will also sponsor the education of the... crew members' children until their bachelor degrees."
On Monday, grieving relatives gathered on the seashore in Bali to pay their respects to the crew.
"We have already given our son to the government. Now that he has fallen in this operation, we hope the government will return his remains to us after all the official ceremonies," said Wayan Darmanta, the uncle of submariner I Gede Kartika.
The KRI Nanggala went missing after requesting permission to dive. The German-built vessel was more than 40 years old but the navy says it underwent a refit in 2012.
The disappearance led to an international search operation with US, Australia, Singapore, Malaysia, and India providing help. It had been hoped that the submarine's hull would be intact and the crew could be rescued.
But on Sunday officials said they had received signals from the sub's location and an underwater rescue vehicle loaned by Singapore was sent down to get visual confirmation of the wreckage.
The depth of the wreck was well beyond the sub's survivable limit.