Norah Jones, Royal Festival Hall

It's easy to forget exactly how successful Norah Jones is, but with over 50 million records sold, she is a modern success up there with the Jay-Zs of this world. To see her come on stage last night, though, you wouldn't have known it. There were no fireworks, no build-up of drama, no crazed intro tape, no MC on stage to announce her entrance, just a band and singer walking on stage to play.

Her young band, it became clear from the outset, were brilliant. Looking like shy boys plucked from a school battle of the bands competition in some Louisiana backwater in 1974 – or, if you prefer, like the Kings Of Leon minus the ghastly arrogance – they kicked up gentle grooves for tracks from Little Broken Hearts full of space that lesser musicians would have filled. They sat back on everything so expertly that it almost felt like no one person was keeping the beat and that they were all floating along as part of some natural process. And Jones, moving between Fender Rhodes, upright piano and guitar, was a part of that band, and locked into that just-behind beat.

Jones's deceptively simple melodies found uncharted lines through the chord patterns

Everything was about restraint, about not messing with the country-soul formula – but the more they continued the more the expertise this took became apparent. And by the time of “Black” from the Rome album by Danger Mouse which Jones appeared on, the flexibility of the formula became apparent too, a cosmopolitan mid-20th-century sophistication emerging from the down-home harmonics. And as the bassist picked up an upright bass and they swung into Hank Williams's “Cold, Cold Heart” they made very clear just how sophisticated country music itself really is.

It didn't seem out of place, though: Jones's new songs felt just as much like standards as did Williams's classic. Everything, every line and every riff, was a hook, just as in high energy radio pop songs, but with the smoky, musty tang of well-worn classics running through them. There was a drasticness about the performance, with the controlled purity of Jones's voice at the heart of it. The power of the performance brought the very human darkness out of every song, particularly in a solo piano delivery of her breakthrough hit “Don't Know Why”, with every layer of meaning in the central phrase “don't know why I didn't come” pulled apart and held open to view not by histrionics but by implacable repetition.

An encore in which the band gathered around the centre of the stage with acoustic instruments and just a single microphone in the middle, was really quite beautiful. But what would have looked like a Mumfordian protestation of retro-ist credentials in lesser hands just seemed natural in this show. There was next to nothing in the entire set that couldn't, in theory, have been done 40 or more years ago, but it still felt new because the songs themselves were fresh - Jones's deceptively simple melodies found uncharted lines through the chord patterns - and the delivery was done by musicians who very obviously played for pleasure in sound rather than to resurrect something dead. With all those sales, Norah Jones can pretty much do what she wants; and on the evidence of last night, what she really wants to do is play. To see a musician so successful still so dedicated to their craft is always a joy, and this show was an immense joy through and through.