It has been more than 20 years since President George W. Bush looked into Vladimir Putin’s eyes and thought he spotted a soul. As various developments have suggested, this may have been wishful thinking; politicians and pundits have tried to tap into the Russian president’s psyche, and have asked themselves how he has managed to sustain power and even popularity at home.

I would advise them all to rush to the Ethel Barrymore Theatre, where Peter Morgan’s “Patriots” is offering one of the most absorbing accounts of political intrigue I’ve ever seen on a Broadway stage. Mr. Morgan is best known to TV audiences for creating “The Crown,” inspired by his 2013 play “The Audience,” which depicted meetings between Queen Elizabeth II and her numerous prime ministers; he also examined a controversial American president and British broadcaster in the play “Frost/Nixon,” which he adapted for a well-received film.

Mr. Morgan has found his richest subjects yet, though, in the man who has led Russia since 2000 (with one four-year gap) and another who tried, in vain, to oppose him: the late oligarch Boris Berezovsky, who as a government official was integral in Mr. Putin’s rise before conflicting ambitions — for themselves as much as their country — turned them against each other.

First presented at London’s Almeida Theatre two years ago, “Patriots” is directed by Rupert Goold, who in previous productions on Broadway and in the U.K. has brought flashy theatricality to studies of notorious leaders ranging from Macbeth to Rupert Murdoch. Here, Mr. Goold’s collaborators include Miriam Buether, whose vast, stark set either looms eerily or becomes a crash pad for blasts of music and noise — both by sound designer Adam Cork — and for Ash J. Woodward’s sometimes harrowing video and projection design. 

Yet “Patriots” is a pretty naturalistic work, notwithstanding the flashbacks that take us to Boris’s youth and key periods in his life, when his choices and those of others prove crucial. The word “decision” appears more than 20 times in Mr. Morgan’s text; we learn that Boris, a math prodigy in childhood, focused his doctoral thesis on decision-making theory, and through his rise and business and politics — the latter occurs under Mr. Putin’s predecessor, Boris Yeltsin — he has prided himself on his ability to find and secure solutions that serve his interests.

In providing key early support for Vladimir Putin, though, Boris Berezovsky sets the stage for his own ruin. When Boris meets Vladimir here, the latter is deputy mayor of St. Petersburg; earnest to the point of being square, he’s also a straight arrow, shooting down Boris’s offers of luxury automobiles. “I don’t accept bribes,” the future president says flatly, adding that his more modest car has “sentimental value. It used to belong to my parents.”

In a quietly captivating performance, Will Keen, reprising his role in the Almeida and West End productions, makes the ascendant Vladimir credible as a man who loves his family and country — and whose apparent humility and righteousness would have appealed to many Russians during a time of economic and political crisis and rampant corruption. 

Boris is his foil in style and substance: a loud, crass guy who cheats on his wife, and is patently eager to exploit the financial turmoil surrounding him to his own advantage. Michael Stuhlbarg, new to the cast, brings his characteristic dynamism to the part; his Boris is by turns repulsive and tragic, turning to denial and then desperation as Mr. Keen’s Vladimir, while more or less keeping his cool, grows increasingly monstrous.

The superb cast also includes Luke Thallon, who wields an Elon Musk-like mix of awkwardness and cunning as Roman Abramovich, one of the oligarchs who has proved able to thrive under Mr. Putin; and a sturdy, searing Alex Hurt as Alexander Litvinenko, the Russian security officer-turned-defector who famously paid the ultimate price for falling out of the president’s favor.

An air of suspicion continues to linger over Berezovsky’s own death 11 years ago, which Messrs. Morgan and Goold reference in a stunning, chilling final passage. By this point, our hearts have gone out to this unlikable man — and they are breaking for Russia.