Stars of stage and screen tend to fall foul of critical prejudice when they make music. Now Scarlett Johanssen has teamed up with TV On The Radio's David Andrew Sitek to record an album of songs by Tom Waits – and it works.

When Tom Waits started his musical apprenticeship in the LA of the early 1970s, he harbored secret Tin Pan Alley fantasies of having his work recorded by the likes of Ray Charles. Though the latter never did get around to cutting "San Diego Serenade," dozens of artists have interpreted Waits' songs over the intervening years. A few have even dedicated whole albums to the man's works.

Holly Cole's quirky Temptation (1995) spanned his 20-year oeuvre from "I Want You" to "The Briar and the Rose"; John Hammond's earthier Wicked Grin (2000) – produced by Waits himself – wrestled with mutant blues classics like "Heartattack and Vine" and "Gin-Soaked Boy". Now along comes actress Scarlett Johansson with Anywhere I Lay My Head. "I read something about that in the newspaper," Waits said on hearing of the Lost in Translation star's plans to record his songs. "I guess that's cool, I don't know. I wasn't aware that she sang."

While Wicked Grin and Anywhere I Lay My Head have a song in common – Orphans' "Fannin Street" – any similarity between the two albums ends there. Indeed, a less Waitsian album of Tom Waits songs than Johansson's would be hard to imagine. Where Hammond's "Fannin Street" was true to the song's Leadbelly-infused spirit, Johansson's sounds weirdly like Joy Division's "Atmosphere". It also boasts David Bowie on backing vocals and the Yeah Yeah Yeahs' Nick Zinner on slide guitar.

Unsurprisingly, Waits and his wife/songwriting partner Kathleen Brennan have given Anywhere I Lay My Head their blessing, confirming as it does that he is the hippest old fart on Planet Pop. They will have been pleased, too, that the album contains just one example of Waits' work prior to his meeting Brennan in 1980 – a spectrally lovely version of 1976's "I Wish I Was in New Orleans". (The choice of that song is incidentally rather apt. For though the album was undoubtedly conceived at some painfully hip party in Williamsburg, it was recorded in an out-of-the-way studio in rural Louisiana.)

Waits and Brennan will have been intrigued by the treatment accorded such eccentric choices as Swordfishtrombones' "Town With No Cheer" and Real Gone's "Green Grass". After all, what would have been the point in doing an album of Tom Waits songs in a slavishly Waitsian style?

Supposedly Johansson, in partnership with TV On The Radio's David Andrew Sitek, started out in more faithful lo-fi style before opting to filter the ten Waits songs (and a sole original, the haunting "Song for Jo") through an indie-rock sensibility that's equal parts postpunk-gothic, 4AD dreampop, shoegazing drone, and TVOTR epicness. It's an approach that renders many of the songs all but unrecognisable. Sitek's sound is the opposite of Waits'. Where the latter is grainy and in-your-face, Anywhere is surging, wavelike, vaporous – "like Tinkerbell on cough syrup," Sitek says.

Big Time's "Falling Down" becomes a fusion of TVOTR and Sigur Ros, topped off with a mini-hommage to the Cocteau Twins. The imprint of Joy Division producer Martin Hannett is all over the album, as is the influence of Suicide ballads like "Cheree" and "Dream Baby Dream". There's even a cheeky drum-machine rendition of Bone Machine's "I Don't Want to Grow Up" that sounds like a Madonna demo from 1982.

So can Johansson actually sing? Her blankly androgynous alto timbre is hardly distinctive but that barely matters. At moments she's a Vanity Fair Nico; at others she merely has an aural crush on sometime Moldy Peach Kimya Dawson. In the end Anywhere I Lay My Head reminds one of a hundred tribute albums – not least Step Right Up, the 1995 album of pre-Brennan Waits songs done by second-division indie bands like Frente! and Drugstore. But it's a bravely eccentric selection and a captivating homage to a singular singer-songwriter.

Barney Hoskyns' biography of Tom Waits will be published by Faber in 2009.

Read Tom Waits interviews and reviews at www.rocksbackpages.com. Over 12,000 articles by the greatest writers from the finest rock publications of the last 40 years.