Pushkin was 26 when he wrote Boris Godunov, the same age as Shakespeare when he chronicled English history in Henry VI. And watching Declan Donnellan's exhilarating Russian revival of Pushkin's 1825 epic, one is reminded not just of its contemporary relevance, but its Shakespearean debt. When the guilt-ridden Boris wrestles with his own murderous conscience, we are straight into Macbeth.

Everything about this production brings Pushkin's play home to us. It is played on a 20-metre catwalk that bisects the Brighton Festival audience. It is also done in modern dress. As a result, the story of how Boris's dubious claim to the Russian throne is challenged by a young pretender – a monk passing himself off as the son of the previous tsar – acquires a fierce topicality. We get power battles in Moscow, trouble on the Lithuanian border, nightly TV bulletins and a sense that the Russian people are victims of cyclical tyrannies.

Donnellan is not the first director to see the play's modern relevance: Yury Petrovich Lyubimov's production, which came to Edinburgh in 1989, treated the play as an anti-Soviet parable. But I suspect, for a British audience, the Shakespearean echoes will emerge even more strongly. Pushkin loved Shakespeare because of the "variegated and multiple" nature of his characters. Accordingly, Boris is both a murdering usurper and an introspective hero, while the monk, Grigory, is simultaneously a deranged imposter and a focus of national discontent. As he himself says of his followers: "They don't care if I'm Dmitry – I'm a pretext for discords and wars."

Instead of a romantic tragedy we get here a wittily allusive play staged with breathtaking panache – nowhere better than the scene in which the militaristic monk woos Marina, daughter of a Polish nobleman. As they slowly strip each other and leap into a standing pool, it is stunningly erotic; but it also becomes a study in the aphrodisiac of power as Marina is drawn to Grigory by his tsarist claims, recoils once she learns he is an imposter and then succumbs to his grandiose idea of himself. Throughout, Donnellan suggests that in an oppressive society people treat the self-invented hero as a symbol of liberation: at one point Grigory is showered with gifts of vodka and sausage by his readily hoodwinked followers.

Aided by surtitles and expressive body language, a handpicked Russian cast brilliantly transcends the language barrier. Aleksandr Feklistov's Boris is a besuited, nervously smoking figure who embodies the guilt of unearned power. Yevgeny Mironov's Grigory becomes a neurotic shape-shifter believing in his own sense of historical destiny. And Irina Grineva's Streisand-profiled Marina highlights the equation between sex and power. What we get is a thrilling vision of Pushkin's young masterpiece constantly illumined by Shakespeare's presence.