A D.J. pawing through a carton of old LPs — Natalie Cole, Angela Bofill — comes upon a curiosity: the original cast album of “Cats.” When he opens the gatefold, glittery spangles fly everywhere.

That’s how “Cats: The Jellicle Ball” begins, and it’s basically what the Perelman Performing Arts Center’s drag remake of the Broadway behemoth does to the drab original. It sets the joy free.

Whether upper- or lowercase, cats never previously offered me much pleasure. The underlying T.S. Eliot poems, ad libbed for his godchildren, are agreeable piffle, hardly up there with “Prufrock” as fodder for the ages. The musical, instead of honoring the material’s delicacy, stomped all over it, leaving heavy mud prints. Andrew Lloyd Webber’s score, and especially the rigged-up story and original staging by Trevor Nunn, tried so hard to make big statements from little ditties and kitties that it wound up a perfect example of camp.

Camp, cleverly, is the new version’s base line, neutralizing that criticism. It turns out that the show once advertised vaguely (and threateningly) as “now and forever” — it ran on Broadway from 1982 to 2000 — works far better in a specific past.

That past is the world of drag balls, which at the time of the original “Cats” was beginning to achieve mainstream awareness. Madonna’s appropriation of the participants’ style and dance moves in her videos and concerts, as well as Jennie Livingston’s celebration of them in her documentary “Paris Is Burning,” helped pave the way for the supremacy of RuPaul and dragmania today. But beneath that triumph lay a darker truth: that the thrill of ball culture depended on its drawing extravagance from destitution, meeting prejudice with bravery, and staring down death with style.

The key insight of this “Jellicle Ball,” which opened on Thursday at the new downtown arts cube, is that at least some of those themes could resonate with Eliot’s subtext and Lloyd Webber’s score. The directors Zhailon Levingston and Bill Rauch have thus transported Grizabella, Skimbleshanks, Rum Tum Tugger and the rest from a metaphysical junkyard to a hotel ballroom for a vogueing competition, accompanied by new versions of the songs that go heavier on the synthesizers, turn some lyrics into raps and add a distinctive house beat.

It’s often a good fit. The former felines — now fantastically attired humans — compete in traditional categories, like Opulence and Hair Affair, that are to some degree matched to Eliot’s descriptions. The song “Mungojerrie and Rumpleteazer,” for instance, pits those two “knockabout clowns” against the pairing of the balletic Victoria and the acrobatic Tumblebrutus in a showdown called Tag Team Performance.

Not that it is any easier to keep the cats straight just because they’re queer. The structure of the show doesn’t allow it. Hemmed in by the Eliot estate, Nunn could not add dialogue, making it difficult to flesh out any characters or encourage specific emotional investment. His solution was a bizarre framing device with late-1970s woo-woo overtones: The clan meets each year on the evening of the Jellicle moon so that their leader, Old Deuteronomy, can choose one lucky cat to ascend to the Heavyside Layer and be reborn.

And if “Jellicle Ball” doesn’t quite solve that problem, it succeeds in making it mostly irrelevant. The new frame allows you to feel something for the characters, at least as a group, even when you don’t know what’s going on, which is often. The design of the long, narrow room, with the audience surrounding a runway on three sides, is awkward in the way one imagines the balls were: You can’t see everything, you’re constantly craning, the sound (by Kai Harada) is blurry and some fuss or hilarity is always happening somewhere you missed.

Even so, we recognize Rum Tum Tugger (Sydney James Harcourt) far better now that he competes in the Realness and Body competitions. (He’s a smooth playah.) Gus, the theater cat, is a more instantly recognizable type as performed by Junior LaBeija, the M.C. of the “Paris Is Burning” ball, as a catty old queen who, though “no longer a terror” can still throw ample shade. And it takes little more than the arrival of André De Shields, with his unsurpassed ability to freeze attention onstage, to show us that Old Deuteronomy is a Moses.

It helps, too, that he’s given a glowing Ten Commandments-like set of tablets, and that he’s dressed (by Qween Jean) in royal purple topped by a gigantic matching lion’s mane (by Nikiya Mathis). Indeed, the wonderfully over-the-top design of the show is as important as the concept itself in filling out the vast blanks of the characters as written. Enjoyable as that is in itself, the chief benefit of the physical staging (on sets by Rachel Hauck, with lighting by Adam Honoré and projections by Brittany Bland) is that it grounds the performative mayhem on the runway in a real environment that suggests the struggles of real lives.

Among other things, this rescues the nominal star role, Grizabella, from bathos. A faded “glamour cat” seeking the reincarnation nod, she has no other function in the original story, not even suspense. (We know she’s going to be chosen because she keeps popping up to sing fragments of “Memory.”) But here, in smeary makeup, a ratty fur and carrying a tarnished old trophy, scrambling about the outskirts of the action, we see at a glance the pain of an outsider now exiled from the place she’d once been safe. Especially as played by Chasity Moore, known in the ball world as Tempress, that pain feels authentic.

That is not something that ever occurred to me in watching the old-school “Cats.” At best the Broadway show felt like a stoned oratorio about nothing, with a dog’s breakfast of song styles including ear-wormy music hall, grating electronica and the occasional Gilbert and Sullivan chorale. (The choral singing here, under the direction of William Waldrop, is gorgeous.) Likewise, the original choreography, by the Royal Ballet star Gillian Lynne, seemed totally random despite its supposedly catlike footwork. The athletic vogueing created for this production by Arturo Lyons and Omari Wiles, sometime blended with throwbacks to Lynne’s classical style, is instead perfectly tailored to its milieu, and thrilling besides.

I should say at this point that, no, I haven’t turned into a fan of the show itself, the one you can see at your community theater or license for your high school. I don’t believe musicals should need whisker consultants. But as happens occasionally, the right idea can transform the wrong material. If “Cats: The Jellicle Ball” has managed a Grizabella turn, reincarnating itself in fabulousness, do not expect an 18-year run or, pardon me, copycat productions. It’s a lightning strike: not now and forever but now and once.

Cats: The Jellicle Ball
Through July 28 at the Perelman Arts Center, Manhattan; pacnyc.org. Running time: 2 hours 30 minutes.