Tying a video Russian-born video game’s sale to the West to the fall of the U.S.S.R. may be a bit of a reach. But “Tetris,” the thoroughly engrossing new film about that, lacks nothing for ambition.

But “Tetris” becomes an equal parts playful and suspenseful yarn in the hands of the director of “Filth” and “Stan & Ollie,” Jon S. Baird.

Computer programmers and commissars, Japanese hardware icons and a British media baron all take the stage in the story about the unlikely path the Gameboy-popular building-blocks game took from Russian inventor to Nintendo blockbuster.

Taron Egerton stars as Henk Rogers, the Dutch-born, New York-raised, Tokyo-based game builder whose Bullet Proof Software gamble on a digital version of the Asian game “Go” is all but forgotten the moment he sees Tetris demonstrated at the booth next to his at the ’88 Vegas Consumer Electronics Show.

He wants rights, but most are already sold. Not in Japan, though, and that’s where he’s based. His pursuit of PC and arcade rights to Tetris and to get Nintendo to commit to “partnering” for it are how this journey down the rabbit hole begins.

There’s this Hungarian/British tech businessman Robert Stein (Toby Jones, in rare form) who hunts for games in the U.S.S.R., and who discovers one that has been shared by everybody who has a computer there. He’s got the rights to it, and with the backing of Soviet leader Gorbachev’s multi-media billionaire “friend,” Robert Maxwell (Roger Allam, vocally, temperamentally and prosthetically perfect), he figures the world is his oyster.

Japanese rights? Sure. Go ahead.

But Rogers’ mesmerizing sales pitch to Nintendo’s founder, Hiroshi Yamauchi (Togo Igawa), all about how “partners are what make us great,” because that’s “why Mario (Super Mario Brothers) has Luigi,” may all be for naught.

The Russians have little idea of what’s become of this game, what contracts have been signed, what’s valid and how to cope with these damned eager beaver Western and Far Eastern capitalists. The inventor of the game, Alexey Pajitnov (Nikita Efremov)? He’s been cut out of the conversation altogether.

As it’s the very late ’80s, the U.S.S.R. is just entering its death spiral, with some officials dogmatically toeing the Communist Party all-for-the-state line and ready to drive a bargain, and others looking for bribes because the End if Nigh, why not fly there — illegally — and try to negotiate in-person with people who have spent decades making Westerners they don’t like disappear?

Egerton manages a fine American accent and even tosses Japanese in as we see what this all-or-nothing gamble is doing to his wife (Ayane), family and Tokyo home life. He’s perfectly cast as a born salesman with the pluck to risk hearing direct threats from assorted Russians and yet persist. Rogers is that desperate.

“I am not going home without a deal.”

The Soviet/Russian skullduggery sequences, with game-inventor Pajitnov straining to keep his head down lest he lose it, are menacing and maddening. Each concerned party is being played off against the rest by mistrustful Russians who are the least trustworthy of all.

Baird inserts digital block-ish Gameboy-styled graphics into random scenes — a building about to be visited, a car chase — to playful effect. The score has “bleeps” and “beeps” tucked into it in between Russian-language versions of the pop radio hits of the day — “Heart of Glass” among them.

The portrayal of the Maxwells — by Allan and Anthony Boyle as the prickly and insecure heir taking charge of this negotiation, Kevin Maxwell — are just delicious. We may know what’s coming, remembering the downfall of the Hungarian-born British media baron and just-as-right-wing rival to Rupert Murdoch. But we can savor it as his company’s digital game division scrambles to score a big win even as the cash has run out.

“Tetris” suffers somewhat from a complicated plot rendered in broad strokes and a story told kind of piecemeal. The opening act is Rogers explaining how Tetris is about to be the next big thing and how negotiations are going so far to a skeptical banker (Nicky Yune). That framing device falls by the wayside as trips are taken, deals are struck and then reneged on, seemingly by everyone but Rogers.

Screenwriter Noah Pink, who wrote the docu-drama series “Genius” for National Geographic, wrestles with a complex and convoluted story and manages to make the murk as clear as it probably can be made. It was always going to be hard to follow all this, and how it ties in to the Fall of Berlin Wall.

But the players, the stakes and the milieu make “Tetris” well worth your time, especially for anyone nostalgic for all the time we wasted on this simple yet elementally addicting game.

Rating: R, beatings, profanity

Cast: Taron Egerton, Nikita Efremov, Ayane, Nicky Yune, Toby Jones, Olag Stefan, Toyo Nagawa, Anthony Boyle and Roger Allam

Credits: Directed by Jon S. Baird, scripted by Noah Pink. An Apple TV+ release.

Running time: 1:58