Grizzly Bear don't have lavish videos featuring big cars and girls, expensively produced songs about bling or even an easily recognisable frontman. However, none of this has stopped Radiohead's Jonny Greenwood pronouncing them his favourite band or the Brooklyners taking a place in the UK top 20 and US top 10.

The band seem as surprised as anybody that, after a decade, the world is suddenly listening in such numbers to 2009's Veckatimest or the new, splendourous Shields. "I'm not gonna lie," begins main singer Edward Droste, surveying the largish Academy as if it were Wembley stadium. "I'm excited to be here."

This is as close as the singer, who is terrible at stage banter, gets to the conventions of crowd-pleasing. Grizzly Bear are almost comically immobile performers, and there are cheers of delight – or relief – when the stage suddenly glows with what look like giant, suspended Halloween lanterns. But it's the songs that are the stars here.

Nobody makes music quite like this: beautiful tunes created from guitar virtuosity, unusual time signatures, post-punk angles, pastoral prog rock, Daniel Rossen's swooping, Jeff Buckley-like falsetto and what the heartfelt-voiced Droste accurately pinpoints in What's Wrong as "a mounting wave of sound". The stillness on stage has the uncanny effect of emphasising the motion and drama in the music: the way A Simple Answer erupts like a fairground stomping Joe Meek classic, or Half Gate and Speak in Rounds gather momentum like galloping horses.

A mesmerising, two-hour set never once loses the attention, and even the spotlight-shy Droste is moved into leading the handclaps. When the stage blazes with yellow for the triumph of Sun in Your Eyes, it feels as if these unlikely superstars do understand showbiz after all.

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