Janelle Monáe, Shepherd's Bush Empire, London

Janelle Monáe piles her theatrical conceits high. A top-hatted, be-whiskered gent comes on stage to introduce "the emotion picture show, The ArchAndroid" – The ArchAndroid being the title of Monáe's stunning debut album. But before we could settle in, the pre-filmed "picture show" turns out to be a sci-fi story of a time-travelling android sent from the future to save us. Its only commandment? "You will dance or die."

"Dance or Die" is the second track on The ArchAndroid, and as it begins, one of three hooded figures onstage is revealed to be Monáe. She de-robes, unveiling an enormous quiff sticking up like the comb of a strutting cockerel, and her signature buttoned-up white shirt and black trousers. She raps and body-pops furiously through the track, already exhibiting her eclectic musical approach, throwing in funk, R&B, hip-hop, electropop, diva soul, prog rock... Obeying that "dance or die" threat won't be a chore.

It seems as if we're to enjoy the whole album in order: the third track, "Faster", slides into the fourth, "Locked Inside", both delivered with a hammer-blow of energy from Monáe and her similarly het-up band. And it's not just the running order she keeps: amazingly, Monáe retains all the polish of her on-record vocals, which move from soul standard to musical theatre to punk roar.

Then the order shifts and things get taken down a notch with her cover of an old classic, "Smile", accompanied only by some electric guitar noodling. But Monáe carries off the chanteuse thing: it's an achingly pure, fulsome performance, and the effect is shiver-inducing.

We go back into fantasy territory with "Wondaland" (also the name of the "arts society" of which she's a founding member), which opens with a series of yelps before a cast of madcap characters join her to dance around to the niftily phrased pop tune. "Mushrooms & Roses" has a muddier wall of psyched-out sounds, as Monáe's voice descends to a low growl. As it builds, she also rather foolishly paints an abstract bottom on a large easel – Wondaland may be an arts collective, but let's hope Monáe sticks to the sonic side of things.

After the lovely ballad "Oh, Maker", we get to the hits: "Cold War", one of her most accessible – and gloriously danceable – tracks blends urgent beats with soul vocals so ferocious they make her quiff fall down. Then the band move into the frantically fast-paced funk of "Tightrope". Her feet look like they should be leaving smoke trails, and there's a big brass-tooting and pelvis-thrusting finish.

An encore cover of The Beatles' "I Want to Hold Your Hand" hardly seems an obvious choice – but then, what would? Finally, on "Come Alive (War of the Roses)", Monáe veers between cabaret diva and abrasive shrieking that wouldn't sound out of place at a heavy-metal gig. With invitations to "go wild" and dance "like a schizo", it doesn't have the most PC lyrics – but no one can deny the wildness of Monáe's own dancing (she even fakes a fit, twitching on the stage floor). The ArchAndroid's world is a pretty intense place, but we all happily let Monáe lead us down the rabbit hole.