The 11-second clip was originally recorded in 1966, by a family in Liverpool. They used their own camera to tape the BBC show from their television screen.
The silent 8mm film was unearthed in Mexico last year, where it had been taken by a collector.
Incredibly, it is the only known recording of the Fab Four's TOTP performance, where they mimed Paperback Writer.
"People just thought pop music would go on forever," archive hunter Chris Perry told Sky News.
"There were literally hundreds of appearances by The Beatles, so why keep another one? Tapes were £400 each in 1971, a huge amount of money, so why keep a recording when you could reuse it?"
The clip will be shown this weekend at the British Film Institute's Missing Believed Wiped festival.
The event will also showcase other recently uncovered footage, including a 1971 live performance of Rocket Man by Elton John and a clip of one of Sky TV's first ever programme, Sky-FI, a 1985 music show presented by Tony Blackburn.
Festival coordinator Dick Fiddy told Sky News: "Obviously we've got hundreds of thousands of items in the archive already, but any new bits that turn up just shows you the past in a slightly different light.
"Television is pretty good at holding up a mirror to society at that time, it's a good reflector of what was going on. So it's almost a short-cut to history, it's a window on the past."