There's no doubting the topical relevance of Bertolt Brecht's epic play about the life of the 17th century scientist Galileo, who saw the truth of the way the universe worked but was forced to stay silent in order to preserve the authority of the church and state. Its lessons about the dangers of speaking truth to power, and the desire of societies to keep the powerless and poor in their place are as important today as they ever were.

But gosh, does he go on about it. Like George Bernard Shaw, Brecht can never say something once if 15 times will do. So one of the many virtues of Joe Wright's exuberant and inventive production is that it makes Galileo's discoveries and his defence of them seem exciting – even admitting to the cutting of the occasional scene with a moment of properly Brechtian alienation and rather more modern comedy.

Wright has only rarely worked in theatre after becoming best known for his movies such as Atonement and Anna Karenina and you can almost sense his pleasure in opening the theatrical dressing up box again. Lizzie Clachan's design is a madly eclectic mix of old and new, and puts some of the audience literally in the middle of the action, in a central circle of cushions around which the action unfolds at hectic speed on a wooden walkway. Overhead is a huge circular disc.

Wright grew up at his parents' Little Angel puppet theatre in London, and he makes full use of that background here, introducing a beautiful puppet Galileo to guide us through the scenes (puppet direction Sarah Wright) and huge paper mâché heads and masks. But he adds to traditional methods of storytelling, state of the art projections from 59 Productions that turn the circular disc into a planetarium and project space over our heads, planets spinning dizzyingly into view like huge, exploding flowers. Add music by Tom Rowlands (one half of The Chemical Brothers) and you have all the ingredients of a wonderful extravaganza.

It's a production full of great images, and expressive movement (the movement director is Javier de Frutos). Moments such as the one where the new Pope is dressed in front of us, changing his ideas from liberal to oppressive as the flowing robes of office are shaped around his naked body, or the sinister emergence of Paul Hunter's chief inquisitor, a dark silhouette against blinding white light, are both visually stunning and intellectually satisfying since they embody the meaning of the piece.

It is all rooted in the earth by Brendan Cowell's performance as Galileo. He's a bearded bear of a man, dressed in tatty T-shirt and jeans, bouncing on the balls of his feet with the thrill of science, that tool of reason that can lessen the burden of human existence and open up new vistas of exploration. In the early scenes, when he inspires his pupils and challenges church and state simply to look through his new telescope and see the truth with their own eyes, his charisma sweeps everyone along with him.

This makes the moment when his terror of torture forces him to recant and suppress his discoveries all the more disturbing. By the end, he is a broken figure, slumped in a chair, playing with a child's toy – until he rouses himself for one last act of defiance. It's a terrific, vigorous performance, and my only reservation is that the one disadvantage of the staging is that it means that Cowell constantly has to express himself across a huge space; all conversations are public, with not much sense of intimacy or of the thought behind his pronouncements.

He's surrounded by admirable performances from a small supporting cast, taking many parts with consummate ease. They are all excellent but I particularly liked Billy Howle's eager Andrea, Ayesha Antoine as the landlady Mrs Sarti and Joshua James as Lodovico. Together they convey the arguments of this unwieldy but urgent drama.

Life of Galileo runs at the Young Vic until 1 July.