Girl power exploded at Tuesday’s night’s Taylor Swift concert at L.A.’s Staples Center, the fourth date of a record-breaking five sold-out shows on her “1989” World Tour. Swift was born in 1989, and the album, she told the crowd, “represents a rebirth,” which may or may not have meant something to the roughly 15,000 swooning fans, a large majority of them teens, tweens and even toddlers — though a generous swath of those rocking out were well into their 40s and 50s — sporting glow sticks and fox costumes and pastel mini-dresses lit up like Christmas trees, a sartorial nod to the pink LED-affixed croptop-cum-skirt set Swift wears during her bouncy “How You Get the Girl” number.

But microfashion represents just a fraction of the sway Swift had over concertgoers during her high-octane 2½-hour performance, punctuated by pyrotechnics and grand theatrics, a rotating catwalk, and a satisfying roster of surprise celebrity guests (the presence of Beck and St. Vincent on a live rendition of the former’s “Dreams” might have flummoxed the pint-sized set, but curried enthusiastic favor with anybody over the age of 18; Swift’s glitzy duet with John Legend on “All of Me” inspired squeals of adolescent joy). Opening acts Vance Joy and burgeoning sister-band Haim — the girls’ long ample hair tossed around with wild abandon during feisty guitar licks — set the tone for a “magical” night that made one thing clear: the young really do run the world.

“Thank you so much for caring so much about it,” Swift said, oozing gratitude for its support of “1989,” her fifth studio album and a pop rock departure from the more countrified fare of previous releases “Speak Now” and “Red.” Since its debut in 2014, “1989” has sold over 5.2 million copies.

If there’s such a thing as a pop-star phenomenon, then judging by her Staples run, Swift is it. Debbie Gibson and Tiffany were something of their time, but they had neither the charisma nor the lasting power, and parents who took their kids to see them in concert didn’t come away wanting to buy their records. And Britney Spears had too many problems and too much sex appeal to ever net widespread parental endorsement. But Swift is proof that the musical divide can be bridged if the package is just right. Where others before her seemed manufactured, veritable pop star puppets of the music industry, Swift is of her own invention, chronicling her young adulthood and all its bumps and bruises through lyrics and beats that hit that enviable sweet spot between mass appeal and artistic credibility.

From opening number “Welcome to New York” to chart-toppers “Blank Space,” “Love Story,” “Bad Blood” and “Style,” Swift was putty in the hands of her Staples Center fans — young and old — who sang along to every word while brandishing signs decorated with glitter hearts and staring agog at the singer as she danced onstage, a lanky swan with dewy blue eyes and the gams of a supermodel and just enough gawkiness to render her relatable. She’s polished, but just unpolished enough and, in a way, that is what makes Swift a star. To wit, it wasn’t just the daughters sporting glittery headbands and ringer tees with Swift’s image on the front — it was the moms too. That is Swift: kid tested, parent approved.

“I’m kind of in love with you,” Swift announced before launching into song. “I’m at my happiest when I’m here onstage with you.”

And you believe her. In a world of cynicism and weariness, Swift is the antidote. Not pure exactly — Swift’s willingness to reveal her shortcomings and boy troubles is all part of her commodity — but something verging on promise. If her act is canned — because, let’s face it, these things are rehearsed — then it’s quite possibly the greatest act in pop music history, because nothing about it feels fake. Everything about Swift feels real. She’s as mesmerized by her fans as they are of her.

And if there were those who ever doubted Swift’s sincerity, there were her celebrity girlfriend BFFs on hand to convince one otherwise. Taped sound bites from “besties” Lena Dunham, Selena Gomez, Karlie Kloss and Cara Delevingne played as interstitials between songs, painting a portrait of Swift as a regular girl who likes Chinese food, playing guitar and is, with her bounty of feline pets, in the words of Dunham, “the patron saint of cats.”

Swift’s last song of the night was the monster hit and contagiously catchy girl-power anthem “Shake It Off,” which prompted everybody in the arena to dance like it was their last night on Earth. But as the music ended and Swift exited, it was the lyrics from “Clean,” displayed in white on a stark black screen, that most aptly and poignantly summed up the night: “She lost him but she found herself and somehow that was everything.”