The pithy, one-line character summaries in the opening song of Dave Malloy’s compressed musical of War and Peace take some serious creative liberties with Leo Tolstoy’s four-volume, 15-book, doorstop of a story about love, friendship and life during the Napoleonic wars.

Then again, it saves a lot of time: Natasha is young, Anatole is hot, Pierre is rich and unhappily married. So we are told in the prologue of this loud, heady, modern-yet-period musical, which was staged to acclaim on Broadway in 2016 and is entirely sung through.

Tim Sheader directs this dynamic new production, which has stratospheric levels of energy. It contains all the promise of a new Hamilton or Six in the early scenes as history and high literature are mashed up with club beats, Dr Martens and lashings of irony. That potential is only partly fulfilled because the show is held back by its own polished larkiness though it is hugely and amusingly original all the same.

Malloy does not adapt the book in its entirety but takes 70 pages. The focus is on the fresh-faced Countess Natalya Ilyinichna “Natasha” Rostova (Chumisa Dornford-May). She is engaged to Andrey (Eugene McCoy, niftily playing Andrey’s curmudgeonly father, too) but is seduced by Anatole (Jamie Muscato), while Andrey is on the battlefield. In the background is the dishevelled, plaintive and ruminating Pierre (Declan Bennett), who is stuck in a loveless marriage to Hélène (Cat Simmons, brilliantly cold). Streaking across their skies is the titular comet to give the story its sense of greater meaning.

There is less philosophising, more rock numbers and archness in the early scenes. The stage abounds with emo eyeliner, Jean Paul Gaultier-style kilts, sheer black lace – and that’s just for Tolstoy’s men. Anatole is like a Russian Byron cum New Romantic rock star as he snorts coke with his sister and shimmies up a pole in his hopes to elope with Natasha.

It’s magnificently fun, though it stays too long in an emotionally distant, arch vein, without bringing quite enough depth. There is a “storm of feeling” inside Pierre, he sings, and you wish it would reveal itself sooner in every character.

But there is such visual excitement to this big, brash, punkishly feathered creation that it dazzles even as you wait for the story to reveal its heart. Leslie Travers’ set is bare and industrial, like a backroom at Berghain. Howard Hudson’s lighting design is spectacular in scale and mood, while the empty stage space is creatively filled with Ellen Kane’s choreography.

The singing blows the roof off, with one glittering voice after another; Bennett saturates songs – such as Dust and Ashes – with feeling, as does Natasha’s friend Sonya (Maimuna Memon, gorgeously rich voice) while Dornford-May is a powerhouse in No One Else.

The music is strong, with wavering accordion and violins, along with deep-throated cellos, and the musicians blasting it out seem key to the excitement and drama. But the lyrics do not quite reach the intelligence and wit of the prologue. In Letters, Pierre tells us that in 19th-century Russia “we write letters … we put down in writing what is happening in our lives”. Many of the lyrics are narrative and pedestrian in this way – sometimes for bathetic effect, other times simply flat-footed.

The emotion, when it comes late in the day, brings back a cresting brilliance. It may not quite be up there with Hamilton or Six but this is a terrific creation and at its best it soars.

• At Donmar Warehouse, London, until 8 February