The famous logo, bought by the V&A last week at a US auction, is now being housed permanently at the London museum.
Independent art charity The Art Fund, entirely funded by donations and its 80,000 members, gave 50% towards the total cost of the artwork.
The V&A bought the artwork from its original creator, artist John Pasche.
Pasche produced the pop art design for the Stones in 1970 after Mick Jagger approached the Royal College of Art in London, looking for a design student. The band had reportedly been frustrated by the bland designs offered by their record label Decca Records.
Jagger visited Pasche's degree show and this led to discussions for a logo and other work for the Stones' own label, Rolling Stones Records, after the group's contract ended with Decca Records in 1970.
Pasche was paid just £50 for the logo in 1970 and a further £200 in 1972.
David Barrie, director of The Art Fund, said: "This iconic logo, first used on the Stones' Sticky Fingers album, is one of the most visually dynamic and innovative logos ever created.
"Designed in the UK by a British artist for one of the country's most successful groups of all time, it's wonderful that it has now found a permanent home in London, where the band was originally formed."
Musidor BV, the Rolling Stones' commercial arm, owns the copyright to the logo. The business acquired the rights in 1982.
Victoria Broakes, head of Exhibitions, V&A Theatre and Performance Collections, said: "The Rolling Stones 'Tongue' is one of the first examples of a group using branding and it has become arguably the world's most famous rock logo.
"We are very grateful for the Art Fund's support in helping us acquire this exciting addition to our collections."
Pasche worked with the Rolling Stones from 1970 to 1974 and subsequently with Paul McCartney, The Who, The Stranglers and Dr Feelgood. He was art director at United Artists(Music), Chrysalis Records and the South Bank Centre.