Perhaps it’s easy to forget, nearly 60 years into his career, that the songs of Bob Dylan were made famous by other artists with sweeter, more radio-friendly voices than the one David Bowie later described as a mix of sand and glue. Between 1963 and 1965, Joan Baez, the Byrds, Peter, Paul and Mary and many others all helped turn Dylan into the voice of his generation for people who couldn’t stand his voice.

Eventually his label, CBS, started marketing him with the phrase that “Nobody sings Dylan like Dylan”. Which is still true, even as that untutored yowl – through age, experience and more age – turned into a croon and, finally, a croak. Now, however, he may have a rival to his own title: nobody has ever sung Dylan quite like Nashville-based Australian singer-songwriter Emma Swift. And maybe nobody (other than Dylan) has ever sung him better, either.

Swift’s splendidly titled album Blonde on the Tracks is a collection of eight Dylan songs that she began recording in 2017 and completed earlier this year, when she became the first artist out of the gate to cover I Contain Multitudes, from Dylan’s new album Rough and Rowdy Ways. She sings with clarity and vulnerability, with just a hint of vibrato, but the key to these performances is her firm grasp of Dylan’s phrasing and timing.

That doesn’t mean that Swift doesn’t put her own spin on such well-worn songs as You’re a Big Girl Now. The way she lets her voice drag just a little longer as she sings “I can change, I swear” will drive a corkscrew through your heart. And where Dylan’s band helps him elaborate the gorgeous melody he can now only hint at in I Contain Multitudes, Swift needs no such assistance in a version that preserves the song’s wryness and subtlety.

It helps that Swift’s partner is Robyn Hitchcock, the prolific solo artist who, in the early 1980s, led the English psychedelic punk band the Soft Boys. Hitchcock is a brilliant guitarist who has long drawn on the legacies of both the Byrds and Dylan; his work on Queen Jane Approximately steers the arrangement closer to the folk-rock style of the former than Dylan’s original from Highway 61 Revisited, and eventually opens out into a ringing electric solo.

But the focus always remains on Swift’s voice. For Queen Jane Approximately, drummer Jon Radford sits back on rimshots, never overwhelming the song with his kit. On the verses of One of Us Must Know (Sooner or Later), Swift is so hushed, accompanied by just the barest traces of pedal steel and piano, that it sounds like she’s singing in church – an effect only enhanced when the song’s main refrain swells into a heavenly chorale.

There are a couple of less obvious selections – Going, Going, Gone is from Planet Waves, while The Man in Me is from New Morning, two of Dylan’s relatively neglected early 70s albums, and Swift illuminates both. The bravest is the 12-minute run through Sad Eyed Lady of the Lowlands, the immense closing track from Blonde on Blonde; on Blonde on the Tracks, it’s dead in the album’s centre and never flags, sung with patience and grace.

Swift has had a stop-start career to date, releasing very little music since coming to notice with an acclaimed self-titled EP in 2014. According to Swift, the idea for Blonde on the Tracks was born from a protracted period of depression and writer’s block. And yet it may be the making of her. It’s a beautiful piece of work by a singer with a rare interpretive gift, comparable to Emmylou Harris’s Wrecking Ball in its intent, execution and intimacy.

• Emma Swift’s Blonde on the Tracks is out now through Tiny Ghost and Bandcamp