It’s obviously up against some stiff competition from lingerie adverts and festive albums that came with free Christmas cards, but there’s an argument that Shadows in the Night may be the most improbable moment yet in Bob Dylan’s latterday career. By releasing a collection of standards from the great American songbook, Dylan, presumably inadvertently, joins in a trend begun 14 years ago by Robbie Williams. Ever since Williams proved that you could sell 7m copies of Swing When You’re Winning to an audience who’d never previously evinced much interest in the work of Cole Porter or Johnny Mercer, the great American songbook album has become a kind of sine qua non among rock stars of a certain vintage. They’ve all been at it, from Paul McCartney to Carly Simon to Linda Ronstadt. Rod Stewart seemed to treat the whole business less like a canny career move than a terrible endurance test to inflict on the general public. By the time he released his fifth great American songbook collection, you got the feeling that even the most indefatigable fan of the jazzy standard was on the floor tearfully pleading for mercy, and in danger of developing a nervous twitch brought on by the opening chords of Mack the Knife.

However, Dylan has latterly made a career out of doing the exact opposite of what most of his peers do. They dutifully tour their big hits, or perform classic albums in order; he takes to the stage and either brilliantly reinterprets his back catalogue or wilfully mangles it beyond repair, depending on whether you’re the kind of critic who gets whole paragraphs out of a change of syllabic emphasis in the lyrics of All Along the Watchtower or an audience member who’s heard three-quarters of Like a Rolling Stone without realising it’s Like a Rolling Stone. They make albums that cravenly attempt to conjure up the atmosphere of their best-loved classic works; he makes albums that conjure up a world before Bob Dylan existed – filled with music that sounds like blues or rockabilly or country from an age when pop was as yet untouched by his influence.

The latter is one of the reasons that Shadows in the Night works. Most great American songbook albums feel grafted on to the artist’s career: too obviously glommed together as a money-making exercise or a means of tiding them over when inspiration fails to strike. By contrast, Shadows in the Night sounds entirely of a piece with the albums Dylan has been making for the last decade and a half. Performed by his current touring band and produced by Dylan himself – rather more beautifully than you might expect, given his reputation for bashing everything out in the studio as quickly as possible – it glides languidly along on bowed double bass and waves of pedal steel, occasionally gently supported by pillowy, muted brass. The playing is full of lovely, subtle touches: the guitar line that shivers in the background of Autumn Leaves’ opening lines; the moment three minutes into I’m a Fool to Want You when the music momentarily loses its rhythmic pulse as Dylan sings “I can’t get along without you”, as if it’s on the verge of collapse. If the album in its entirety sounds more monotone in pace than its immediate predecessors – Dylan’s drummer is frequently relegated to occasionally tapping a hi-hat, or banished from the studio entirely – any of its tracks could have been slipped on to Modern Times or Tempest without provoking puzzlement among listeners.

Certainly, the album fits perfectly with what you might call Dylan’s latterday persona, the grizzled old geezer unveiled on 1997’s Time Out of Mind, either sentimental or growling at the world to get off his lawn; “trying to get to heaven,” as the song of the same name put it, “before they close the door”. Whether that’s a part Dylan is playing or an accurate representation of what he’s like in his 70s is a moot point, but the songs on Shadows in the Night have been chosen – usually from less well-thumbed chapters of the great American songbook – to suit the character. Their lyrical tone is usually remorseful and lovelorn – The Night We Called It a Day, What’ll I Do, Full Moon and Empty Arms – and even when it isn’t, it ends up sounding that way because of Dylan’s delivery. His version of Rodgers and Hammerstein’s Some Enchanted Evening takes a song about a burgeoning romance and ferrets out the misery buried in the lyrics. “Fly to her side and make her your own / Or all through your life you may dream all alone,” he sings, but there’s a rueful quality to his voice that undercuts the carpe diem sentiment and a song cautioning the listener not to miss their chances suddenly becomes a song about missed chances.

A lot has been written about the state of Dylan’s voice in recent years, but if any songs suit a ruined voice, they’re those assembled here. Most of their authors were half Dylan’s age when they wrote them, but they sounded much older: everything is suffused with world-weariness and regret. The irony is that Dylan’s vocals on Shadows in the Night sound “better” in the conventional sense than they have in years, presumably because he’s singing softly – crooning, if you will. There’s certainly nothing here that resembles the opening of Tempest’s Pay in Blood, where a combination of rage and whatever havoc has been visited on his larynx over the years left him sounding like the frontman of Autopsy or Disembowelment, and what came out wasn’t words but a terrifying, incomprehensible growl. Still, such things are relative. His voice is still cracked and catarrhal and occasionally ventures wildly off pitch, usually when he tries to hold the songs’ long, dramatic, final notes. It doesn’t matter: it fits, as if the hard-won experience of the lyrics has been etched on his throat.

Dylanologists could doubtless tell you a lot about the relationship between the songs here and his own oeuvre: you suspect they’ll have a field day with the religious overtones of Stay With Me. To say that all seems besides the point isn’t to rubbish their close reading and study, which at its best is genuinely illuminating. It’s merely to suggest that Shadows in the Night works as an unalloyed pleasure, rather than a research project. It may be the most straightforwardly enjoyable album Dylan’s made since Time Out of Mind. He’s an unlikely candidate to join the serried ranks of rock stars tackling standards: appropriately enough, given that Frank Sinatra sang all these songs before him, he does it his way, and to dazzling effect.

• Bob Dylan on Sinatra: ‘Frank is the mountain you have to climb’