We gathered on a freezing Sunday night, inside a barrel-vaulted church designed in the 1890s by Charles Rennie Mackintosh, to witness a cresting wave. Vulture Prince, the third album by the Brooklyn-based Pakistani singer, composer and producer Arooj Aftab, was one of the most accomplished and interesting records of last year. A keening song of loss, dedicated to her late brother, Vulture Prince is almost impossible to pin down. It’s a flood plain of merging musical streams, a genre-phobic blend of jazz, minimalism, Sufi devotional music, acoustic textures and torch song. Sung almost entirely in Urdu, its beauty and import are immediate, its emotional pull universal.

Following two Grammy nominations and a fat wodge of critical acclaim, Aftab’s stock is in the ascendant. She has a new label, a new stylist, and new product for sale. At 36, after years spent at the ‘specialist’ end of the art-music spectrum, the gear shift seems to be causing a degree of cognitive dissonance. ‘There’s a T-shirt,’ she deadpanned, nodding towards the merch stall in the lobby. ‘It has my face on it.’ A beat. ‘The label made me do it.’

Sipping whisky and dressed, in her own words, as ‘a Goth cupcake’, Aftab proved a droll, sombre, quietly queenly presence. ‘Feel free to wild out,’ she said after the gentle undulations of the opening number, ‘Baghon Main’, drew softly to its conclusion. It was a self-deprecating dig against the slow, sad intensity of her music, which is dark, thick stuff, wrapped in a velvety melancholy. Yet the joke didn’t quite land; there was indeed a kind of wildness at large. This was the final night of the Celtic Connections festival, and the mood was politely clamorous. The room fizzed with the excitement of high expectations first being met and then exceeded. Rapt silence was followed at the end of each song by a raucous, rumbling clatter of the kind usually only heard when a senior hack is being banged out of the newsroom for the final time.

The musical mood was both languorous and intense. Aftab performed eight numbers in an 80-minute set, a modest row of lights strung behind her, a replica Willow tearoom Mackintosh chair on display high above her head. The magnificent heart-shaped window, timbered ceiling and general air of sanctity added to the deep, devotional thrall of the music. We perched on unforgiving pews, happy in our discomfort.

Her voice cast a mesmeric spell: rich, full and flawless, free of melismatic ornament or wild ululation. Yet if her vocals were the star attraction, this was very much a trio show, rather than a solo showcase. There were long periods of instrumental back-and-forth while the singer stood impassively or gazed intensely at her two accompanists: Scottish harpist Maeve Gilchrist and Greek double-bass player Petros Klampanis. Aftab called them two ‘bad cats’, and they lived up to their billing. Both were fantastically mischievous, astonishingly agile. Rocking and swaying her instrument, Gilchrist coaxed funky bass riffs, electronic pulses and rippling arpeggios from her harp. Klampanis was equally inventive. Sometimes no strings were required. On ‘Saans Lo’ — so quiet, so desolately pretty — Klampanis whistled gently over the descending melodic line. Both musicians rapped out a polyrhythmic percussive beat on the body of their instruments during ‘Suroor’.

On stage, Aftab’s music is more obviously jazz-orientated, more opened out, than it is on record. At different times there were echoes of the music of Jeff Buckley, Kate Bush’s 50 Words For Snow, Joanna Newsom and John Martyn’s more expansive, pastoral moments, but comparisons ultimately felt redundant. Klampanis switched to piano for ‘Inayaat’, which plumbed depths of almost fathomless sorrow. You didn’t need to understand Urdu to get the gist. ‘Last Night’, the sole song sung in English, featured only a single mantric line: ‘Last night, my beloved was like the moon.’ Another song was described by Aftab as being about ‘departure, intoxication, dislocation. Bad timing.’

The set ended with ‘Mohabbat’, ‘the banger, the hit’ — more drollery. It seemed to hover in the air despite being underpinned by a fiercely rhythmic pulse. For an encore she returned to her first album, Bird Under Water, and ‘Aey Na Balam’. Over spiky, crab-walking jazz, her voice finally ascended to the heights, pulling us from the undertow of darkness towards the light. Outside, fire engines and police cars blared by. Inside, there was a kind of mercy. It felt like one of those nights where those of us in attendance might feel smugly compelled to say in years to come: ‘I was there.’