Our global age of income inequality certainly cannot be guaranteed to add a jolt of relevance to any classic play, but it definitely serves as an illuminating backdrop to Thomas Ostermeier’s stark and penetrating production of “Miss Julie,” August Strindberg’s celebrated 1888 drama about the turbulent war of wills between a well-to-do young woman and her father’s servant.

For his first visit to the Lincoln Center Festival, Mr. Ostermeier has brought a Russian-language production of the play originally staged in 2011 at the Theater of Nations in Moscow. Like his versions of Ibsen’s “A Doll’s House,” “Hedda Gabler” and “An Enemy of the People,” all seen at the Brooklyn Academy of Music, Mr. Ostermeier’s “Miss Julie” takes place in the here and now, in this case the here and now of Vladimir V. Putin’s Russia, where globe-trotting billionaires minted after the collapse of the Soviet Union live in extreme luxury while much of the population merely subsists.

The production, at City Center through Sunday, takes places on a minimalist set depicting a gleaming modern kitchen decked out in stainless steel. Video has become an almost obligatory element of European avant-garde theater (see Declan Donnellan’s stimulating “Ubu Roi,” also at the festival this year), and Mr. Ostermeier’s “Miss Julie” is no exception. A large screen hanging above the black turntable set shows us (mostly) a bird’s-eye view of the proceedings, which begin with the servant Christine (an amusingly unflappable Julia Peresild) meticulously preparing chicken bouillon — for her mistress’s young puppy, who’s been newly spayed.

The time frame has been moved from the original Midsummer’s Eve to New Year’s Eve. Drifts of falling snow pile up around the edges of the stage as the play unfolds, suggesting the frigid Russian winter but also adding a sneaking sense of the suffocating complications that come to oppress the central characters. The other small changes in Mikhail Durnenkov’s vivid new adaptation — Jean (Evgeny Mironov, also the artistic director of the Theater of Nations) is Julie’s father’s chauffeur, not his valet, etc. — smartly align the play with the social climate of contemporary Russia, even as the dialogue and action generally conform to the original. (The play is performed in Russian, with English supertitles.)

Christine and Jean — who are vaguely betrothed, in part at the behest of their boss — exchange gossip about the wayward antics of Julie (Chulpan Khamatova), the daughter of a former general who retired from the army, we eventually learn, and made a fortune. Julie soon flounces into the kitchen looking for entertainment, having tired of dancing with the friends Christine and Jean invited over for a party. They expected to have the place to themselves: Julie was supposed to jet off with her father to the beach, but impulsively withdrew at the last minute.

Wearing a diaphanous wrap dress and towering heels, which she slips on and off as she moves between a flirtation with Jean and sudden assertions of her social superiority, Ms. Khamatova’s already tipsy Julie exudes an air of privileged recklessness, the manner of a young woman who has always had more toys than she knows what to do with. Her feline sensuousness can appear alluring one moment, and predatory the next, as she and Jean are drawn into a gradual intimacy that threatens to destabilize the rigid class divisions between them. (Comparing her to her mother, who similarly liked to consort with the lower orders on occasion, Jean calls her “a democrat — until the wind changes.”)

Mr. Mironov’s Jean doesn’t at first seem the type to assert himself, sexually or otherwise, in the presence of the boss’s daughter. He’s diffident when she impetuously suggests they go join the party and sing a karaoke duet, reminding her that it might be awkward in the morning, when he’s back to driving her to the shopping mall. “I cancel the subordination for today,” she imperiously says, and drags him off.

But Jean has acquired some social graces and experience, including a smattering of English and a taste for good wine, learned when he took a sommelier’s course while working at a hotel. And after he and Julie have slept together (while this is happening, the neon-clad guests invade the kitchen for an impromptu rave), he asserts himself more confidently, believing that having bedded the boss’s daughter he has absorbed some of the power she once had over him.

Mr. Mironov’s tone grows cutting as Jean becomes bold, even contemptuous, toward Julie. At one point during their increasingly heated argument over what to do with their lives, he stuffs her into the freezer and keeps her trapped there for a few minutes.

Ms. Khamatova’s Julie, meanwhile, seems truly disoriented — perhaps for the first time in her life. Dazed confusion distorts her fine features, and soon she’s reaching in desperation for the vodka bottle. When Jean convinces her, at least temporarily, that their only choice is to take flight and start a new life together, she returns onstage decked out in sleek leather pants and a fur-trimmed jacket, her puppy tucked in her ample designer bag. (The puppy, by the way, meets the same unhappy fate as Julie’s bird in the original.)

But young women of her class in contemporary Russia are so removed from ordinary life that this future existence terrifies Julie as much as it entices her. The sense of superiority by which she has lived now seems a strange delusion; not only does she realize that she is, and always has been, devoid of direction; she’s devoid of will. Jean, on the other hand, has been preparing all his life to “crawl out of that pit” he was born into, and if this opportunity doesn’t work out, we get the firm sense that he’ll manufacture another one.

The intriguing idea that Mr. Ostermeier and Mr. Durnenkov’s terrific production teases from this classic text is that the self-insulation of the Russian elite contains the seeds of self-destruction. Then again, as long as there are men like Jean, ambitious, smart and ruthless enough to get their way no matter the cost, the current oligarchs may just be replaced by new ones.