Sonic Youth, Shepherd's Bush Empire, London

Age shall not wither them

Though their reputation was made with avant-garde explorations of rock's atonal fringes, Sonic Youth's current rejuvenation owes far more to their love of rock classicism. This New York institution seemed to be unravelling at the seams with improvised indulgences such as A Thousand Leaves, but the new album, Murray Street, marks their 21st anniversary with their most cohesive set in a decade.

Thurston Moore, a lanky, goofy beanpole, is the nominal master of ceremonies; he and his crew resemble Scooby Doo's chums as they amble on stage. The tousle-haired new recruit Jim O'Rourke is the super-sleuth who has helped to solve their most pressing inquiry: how can a bunch of ageing art rockers transform themselves into a vibrant creative force?

The group are famed for their armoury of guitars and effects. Early reports from their summer-long European tour had O'Rourke on bass, but tonight, underlining their egalitarian set-up, Kim Gordon plays bass while Gentlemen Jim doubles on third guitar and what looks like his own portable mixing-desk.

The perfect foil for the thoughtful and allusive songs Moore composed for Murray Street, which is played tonight in its entirety, O'Rourke seems to have given a shape and focus to the group's wayward style. Before playing "Candle", from their classic 1988 opus Daydream Nation, Moore begs the audience's patience as the band haven't played the song for several years. There's no need: like nearly everything they play, it springs into life with a feast of delicate and dynamic interplay.

The show is gratifyingly at odds with the Youth's sometimes po-faced persona; on the new "Rain on Tin", for instance, their sheer glee is infectious as they uncover melodic sweetness in the song's core.

Moore has long fancied the Youth as latter-day New York inheritors of the Grateful Dead's mantle – a seamless blend of improvisation and tradition. That aspiration is clear tonight. At various points, the songs pay homage to vintage Neil Young, Television, Terry Riley and even John Cage. But, whereas the Dead specialised in expansive dreamscapes, Sonic Youth's compositions are claustrophobic and internalised, making them the quintessential post-11 September rock band.

Like the Dead, they have inspired an alternative nation of followers, most recently the excellent Yeah Yeah Yeahs, a fact underlined by the songs in which Kim Gordon takes centre stage. On the penultimate "Kool Thing", she delivers a call to arms while the boys in the band find myriad ways to orchestrate the sound of controlled destruction. As soon as Moore leaves the stage, he goes to check on his and Gordon's daughter, CoCo. A new Youth generation may be waiting in the wings, but, on this showing, her parents are a long way from retirement.