The album, due out Sept. 23 on Streamline/Columbia/Interscope, represents the latest pop dalliance with the Great American Songbook, something Mr. Bennett, probably more than anyone, knows all about. It also suggests a determinedly classy reboot for Lady Gaga, whose most recent solo release, “Artpop,” fell short of her usual blockbuster standards, delivering no transcendent single on the level of “Bad Romance.” From a distance, the collaboration can look like a tactical maneuver for both artists.

Up close and in person, that suspicion gets softened, if not entirely dispelled, by the affectionate rapport between the singers and by their earnest exaltation of the songbook and its attendant jazz style. “I think it’s just much truer to my nature to sing this way,” Lady Gaga said the morning after the PBS taping.

“I mean, I was telling Tony, ‘Can you imagine if you had the career that you’ve had, but for the first 10 years you couldn’t sing out?’ I wasn’t using my instrument to its full capacity.” Glancing at Mr. Bennett, she added: “I feel set free. I feel let out of a cage.”

It had been a productive morning for the singers, born Anthony Benedetto and Stefani Germanotta, both products of New York City. They’d appeared on NBC’s “Today,” announcing the album and trading encomiums. (Lady Gaga showed off a new tattoo on her biceps, based on a sketch given to her by Mr. Bennett. The image, signed “Benedetto,” was of a trumpet with a mute, à la Miles Davis.)

They were still dressed for television — he in a navy suit, she in a tawny robe and Cleopatra makeup, with a single dark braid over one shoulder — during an hourlong interview at 30 Rockefeller Center, in a dressing room usually reserved for musical guests on “Saturday Night Live.” (Among the glossy photographs arranged on the walls was one bearing this inscription: “SNL ... I’ll take your cheap applause. Lady Gaga.”)

Characterizing her experience with Mr. Bennett, Lady Gaga variously struck the tone of an awed admirer and a grateful peer. “I sit in the back of the theater and watch him, to be inspired and to learn,” she said. “He’s always having this intimate conversation with the whole room. It feels like he’s talking to you, not the person sitting next to you. But every person feels that way. What I do with my music is a little different. I want everyone in the room to kind of feel the same, all at once, together.”

Mr. Bennett, for his part, reiterated the opinion he has publicly held since 2011, when he heard Lady Gaga sing a standard at a charity benefit and promptly invited her to appear on his album “Duets II” (RPM/Columbia). (Their track together — “The Lady Is a Tramp,” inevitably — led off the album, which won a Grammy Award in the traditional pop vocal category.) “The thing I love about being a jazz artist is: It never becomes dated,” Mr. Bennett said. “It’s a wonderful art form, to be spontaneous and just think for the moment. And that’s how she sings. She’s a natural jazz artist.”

Taking words like these to heart, Lady Gaga got choked up a few times during the interview, stopping once to collect her composure. “I wonder where I would be at this moment in my career had this not happened,” she said. “I really didn’t want to make music anymore, for a little while. Because I was so confused and tired. But now it’s so clear.” Mr. Bennett handed her a tissue, and she laughed, wiping away tears.

That someone like Lady Gaga would even aspire to validation as a jazz singer — and “validation” was the word she used — can be understood as the natural byproduct of the last 30 years or so of consensus aspiration in popular music. It was in 1983 that Linda Ronstadt bucked expectations to release a throwback, soignée album titled “What’s New.” Featuring arrangements by Nelson Riddle, best known then as now for his work with Frank Sinatra, it spent weeks in the Top 3, behind Michael Jackson’s “Thriller,” and sold more than three million copies. Stephen Holden, reviewing the album for The New York Times, observed that it “signals a shift in the prevailing winds of pop.”

The next few decades — which brought several sequels from Ms. Ronstadt, along with Natalie Cole’s “Unforgettable” and no fewer than five Rod Stewart albums under a “Great American Songbook” banner — proved that to be at least partly true. Cozying up to standards has become a proven play for all manner of former pop or television stars, eager to prove their savoir faire to a fan base that’s maturing along with them. (Annie Lennox will be the latest to join the club with “Nostalgia,” out on Blue Note on Oct. 21.)

But the precedent more directly applicable to “Cheek to Cheek” was set in 1993, when Frank Sinatra released “Duets.” Produced by Phil Ramone and featuring marquee guests like Luther Vandross, Barbra Streisand and Bono, it gave Sinatra the first multiplatinum album of his storied career.

Mr. Bennett, who also appeared on “Duets,” was by then in the midst of a career resurgence himself. But it was his “MTV Unplugged” appearance the next year that renewed his lease on the pop imagination. He waited a dozen years to make his own duets album, enlisting Mr. Ramone as producer; it won two Grammy Awards and sold more than a million copies. With “Duets II,” released when he was 85, Mr. Bennett became the oldest living artist to reach No. 1 on the Billboard album chart.

Lady Gaga, who has lately made a show of dismissing the arms race of chart rankings and sales figures in pop, was quick to recognize the value of Mr. Bennett’s imprimatur. “I think that if I had done a jazz album without Tony, people would not have taken me seriously,” she said. Which is probably true, even though her singing on “Cheek to Cheek” is strikingly assured, and sometimes outstanding, as in her subtly dramatic versions of the elegant ballads “Lush Life” and “Ev’ry Time We Say Goodbye.”

Elsewhere on the album, which was recorded in a fashion more customary for him than for her — “Live, with the band,” she happily put it, “no Auto-Tune, full takes, no tricks” — the results are uneven. Lady Gaga has sterling vocal control and a feeling for smart embellishment, but she’s so accustomed to riding the front of the beat that her jazz phrasing can feel hokey or stilted. By her own account, she has been singing jazz since the age of 13, but at times, notably on “I Won’t Dance” and “They All Laughed,” her touchstone seems less Ella Fitzgerald than a Broadway trouper like Liza Minnelli.

During the “Great Performances” concert, which will be broadcast in edited form on Oct. 24, the best moments between Mr. Bennett and Lady Gaga were the freest, when any meaningful distance between their styles appeared to dissolve.

Still, there were reminders. Her fans, shouting encouragement between songs, all called her Gaga; Mr. Bennett, in his remarks, gallantly adhered to “Lady.” And an audience that greeted “Anything Goes” with gracious reserve was raucous in its enthusiasm for “Bang Bang (My Baby Shot Me Down),” the 1966 Cher hit, a knockout solo feature for Lady Gaga. (It will be a bonus track on some deluxe editions of the album.)

Then again, there were well-deserved cheers for “But Beautiful,” featuring a lustrous orchestral arrangement, and “It Don’t Mean a Thing (If It Ain’t Got That Swing),” with a guest trumpet solo by Chris Botti. The portion of the room that belonged squarely to Lady Gaga seemed to have little problem getting behind the songs.

“That’s what we’re hoping,” Mr. Bennett said. “That the audience she has will finally listen to the best music that ever came out of the United States.”

Lady Gaga echoed that thought, perhaps with a touch more diplomacy. “I want my fans to understand that this would be like if, 80 years from now, 30 different people had sung ‘Bad Romance,’ and it had been a hit 20 different times,” she said. “I mean, that is a song that has time-traveled, and maneuvered like a sexy little snake through the future.”

She laughed. “People are so obsessed with the future. I admittedly am, too, but more from a cubistic standpoint. I like the idea of nostalgia bringing us a wisdom when we’re traveling forward.”