Crossing the famous Finland station in Leningrad one day in the early 1960s, Kolya Vasin was stopped by a policeman who had spotted his long hair. "You are not a Soviet man!" charged the officer. "And he grabbed my hair," recalls Vasin, who was then hauled across a platform while dozens of people laughed. "I was crying from the pain, but I had to keep silent. I was afraid the man would drag me off to prison."

Vasin was a diehard Beatles fan. The Beatles' music had given him, he said "all the adventures of my life", for which "I was arrested many times, accused of 'breaching social order'. They said anyone who listened to the Beatles was spreading western propaganda." More than that, in the USSR, the Fab Four "were like an integrity test. When anyone said anything against them, we knew just what that person was worth. The authorities, our teachers, even our parents, became idiots to us."

Around this time, in Britain in 1962, a young Russian speaker from Yorkshire called Leslie Woodhead joined Granada TV in Manchester as a junior researcher, whose job included "persuading … local officials or champion knitters" to appear on a programme called People and Places. One week, a show featuring a brass band needed a further item. "There are these kids making a lot of noise in a cellar in Liverpool," advised a fellow researcher. "They haven't made any records yet."

Woodhead duly met them for a drink, and shot the first film of them playing – a lunchtime gig at the Cavern – but transmission was delayed because of a problem with the brass band's union fees. Instead, Woodhead urged his producers to allow the Beatles into Granada's studio, and play on live TV for the first time. They sang Love Me Do and Some Other Guy. Four months later, they reached No 1 with Please Please Me. Despite Woodhead's part in Beatles history, it was not the band's story in north-west England – where he still lives – but in the Soviet Union that became, Woodhead says, "an essential narrative of my times", and that propels his effervescent new book, How the Beatles Rocked the Kremlin.

This tells the remarkable story of precisely how and why, as Woodhead explains, "the Beatles came to mean more, and were more important, to that generation of Soviet youth than they were here, or in America – for several reasons".

The book's main character, the Russian writer and critic Art Troitsky, makes the claim that: "In the big bad west they've had whole huge institutions that spent millions of dollars trying to undermine the Soviet system. And I'm sure the impact of all those stupid cold war institutions has been much, much smaller than the impact of the Beatles."

A grand assertion, maybe – but widely shared. "Beatlemania washed away the foundations of Soviet society," explains Mikhail Safonov at the Institute of Russian History. And the Russian rocker Sasha Lipnitsky – snowflakes falling on his beret as he talks to Woodhead in a park bandstand – insists: "The Beatles brought us the idea of democracy. For many of us, it was the first hole in the iron curtain."

All this became Woodhead's story, too. Before joining Granada, Woodhead had undergone his national service by eavesdropping on radio traffic between Soviet pilots at an airbase near Berlin. He later went on to become the documentary film-maker who, more than any other, recorded – often clandestinely and at risk – the anti-Stalinist underground in eastern Europe, and its eruptions during the late 1980s and early 1990s.

Since then, Woodhead has often travelled through the new Russia to explore the Fab Four's role in the unravelling of a superpower. And of course, among his first ports of call was Kolya Vasin – yellow submarine on the wall of his apartment full of Beatles memorabilia and a cat called Hey Jude.

There are so many others – rock musicians, eccentrics, writers, dissidents – of the same vintage, with different stories to tell, but all variations on the theme. "There was not a band anywhere in the Soviet Union", says Woodhead, "that did not start life as a Beatles tribute band."

The rock musician Boris Grebenshchikov was eventually allowed to cut an album, first with the official Melodiya label, then with CBS in America, after a concert in Leningrad with Dave Stewart of Eurythmics. He speaks of the Beatles with a "mystical musing" that Woodhead says he could not tolerate in any other context. Andrei Makarevich formed a Beatles-inspired band called Time Machine , who became huge in Russia from the 1970s – only to be later denounced as "un-Russian", "advocates of indifference" – and who remain iconic today.

Indeed, the repression and harassment of the music ebbed and flowed as the party controls lapsed or intensified. "It went in waves: sometimes you could be approved for an official recording, and sometimes you were banned, losing your job or education. It must have driven them insane," says Woodhead. He not only excavates the minds of the rebels but also the propaganda machine at work. He recounts how a school staged a mock trial of the Beatles – broadcast on radio – with a prosecutor and denunciations in the manner of Stalin's show trials of the 1930s. A critical bulletin shown on state TV, entitled Pop Quartet the Beatles, told the story of how "these gifted guys could be real cash earners" while, "struck down with psychosis, the fans don't hear anything any more. Hysterics, screams, people fainting!" So ran the TV commentary, accompanied by shots of dancing fans intercut with images of the Ku Klux Klan and dire poverty in the American south. "Keep on dancing, lads, don't look around," the programme taunted, "You don't really want to know what's happening. Keep going, louder and faster! You don't care about anyone else."

As Woodhead points out, to Beatles fans in 1970s Russia, "Everything west was good. The kids came to believe the exact opposite of everything they were being told all those years. Whatever the authorities said was terrible was bound to be wonderful."

Moreover, Woodhead says: "Once people heard the Beatles' wonderful music, it just didn't fit. The authorities' prognosis didn't correspond to what they were listening to. The system was built on fear and lies, and in this way the Beatles put an end to the fear, and exposed the lies."

"The more the state persecuted the Beatles," concurs Mikhail Safonov, "the more they exposed the falsehood and hypocrisy of Soviet ideology."

Looking through the other end of the telescope, it is enlightening to find what the Soviet authorities approved of. They "positively encouraged" disco music – the Bee Gees' Saturday Night Fever, Abba and Boney M (though Rasputin was officially banned) – because, says Woodhead, "it was musically rigid and could be contained within the dance floor, it wasn't going to spill out on to the streets".

A concert by Santana and Joan Baez was cancelled, leading to what Russian history calls the "Rock Riot", crowds dispersed with water cannon and smoke grenades. But, writes Woodhead: "The culture commissars were untroubled by Elton John's Song Book." At Boris Grebenshchikov's concert of 1988, however, Woodhead observes how, "looking out over the kids from the best seats set above the crowd, officials and party bosses sat stiff and uneasy, spectators at a revolution they could not control".

Among Woodhead's themes is that, unlike the Beatles themselves, their insurgent followers in the USSR came from families of the cultural and even political elite: Makarevich's father was a respected architect permitted to travel to the west; Lipnitsky's grandfather interpreted meetings between Nikita Khrushchev and John F Kennedy; one rocker called Stas Namin was the grandson of a former prime minister, and friend of the great composer Dmitri Shostakovich.

The Beatles' imprint on even post-communist Russia is deep and enduring – a punk band called the Oz belts out Working Class Hero and Crippled Inside. And Woodhead's story is woven through with the ironies of "liberation" from communism; at a deeper level this is a book about all rock'n'roll – protest and pop, indeed – not just the Beatles in Russia.

At first, Soviet fans tried to cope as their home-grown but Beatles-inspired idols were tempted, argues Woodhead, by co-operation with the state – only to watch them try to assert themselves in a capitalist west that was entirely indifferent to their work. "Boris Grebenshchikov toured America, and he was a complete non-event there," says Woodhead, "after which, it took him years to recover the esteem of his Russian fans. But then, we in the west are completely unaware of this history. It doesn't help that a bunch of Soviets are singing in Russian what we think of as our music – but there's obviously a great deal of cultural arrogance on our part."

More serious was the eagerness with which capitalism devoured – and was devoured by – Russian society: Woodhead describes Paul McCartney's concert in Kiev, sponsored by an oligarch colossus, just as his famous performance in Red Square, Moscow (at which Vladimir Putin chatted with Makarevich of Time Machine) had been promoted by Alfa-Bank. "Wasn't that a perfect 21st-century deal between rock, money and politics?" writes Woodhead.

In Kiev, he sees crowds shelter from a downpour under Coca-Cola umbrellas and girls on stilt heels flocking to hear McCartney via the shopping mall "in pursuit of pink fripperies". Then, "surrounded by heavy security guards", he reflects, in our conversation, "I found myself asking, is this the Russia these kids inherited from those utopian expectations? Well, yes it is; it opened the Pandora's box."

"I used to struggle against the cops," laments Kolya Vasin, "now I struggle with these fools who do business and worship the dollar."

Then there was that realisation that the west was not the opulent land of stretch limousines it was presumed to be. Some of Woodhead's cast know this; one even has a copy of Back in the DHSS by another Liverpool band, Half Man Half Biscuit. Makarevich was surprised and appalled to find, upon finally making his pilgrimage to Merseyside, that it was so "small and poor". "We were, and still are, exponentially wealthier than they," says Woodhead. "But when did they realise that we're as fucked as they are? Not until after the end of communism."

Why the Beatles? There is no hint of the Rolling Stones or the Who in all this. In Czechoslovakia, the underground was being inspired by dark dissonance in the Velvet Underground and Frank Zappa. "I think the Czechs had that recent memory of democracy, before the war," reflects Woodhead. "And their culture has roots in Kafka and the surreal. But Soviet taste was more melodic, they like tunes above all, even a little sentiment, verging on the beautiful – and there, I'm describing a McCartney song, not hypersexual rock'n'roll, or Street Fighting Man.

"It was also the right music at the right time. There had been this moment of Gagarin in space, the possibility that the Soviets may even win the cold war. Then it just fell to bits, and in the fear and disappointment, and as they said themselves: they 'needed the vitamins', and the vitamins were provided by the Beatles' music."