Andrei Zviagintsev’s prize-winning film tells a story of pervasive corruption and repression that Moscow doesn’t want the world to see.

MOSCOW—Last week, many Russians watched the online, pirated versions of “Leviathan,” the Golden Globe winner this year for best foreign-language film. Arguments broke out, one after another, in a shocking wave of public reaction to a piece of art. Internet forums buzzed in emotional discussions filled with bitter criticism.

When director Andrei Zviagintsev returned from the Golden Globe ceremony, he heard from his countrymen that his movie was full of “rotten lies” and “filthy libel”—that the film should be banned, actors punished, and that a new law must be adopted to protect Russian audiences from movies harmful to the country’s reputation.

Some “Leviathan” haters even hinted that the movie was ordered up by Washington, eager to defame Russia, in a manner similar to the way the CIA used the novel Doctor Zhivago as a literary weapon during the Cold War.

None of the Kremlin’s top officials called to congratulate Zviagintsev on his movie’s victories at Cannes (best screenplay), the Golden Globes, or for its Oscar nomination.

On Monday, officials in the region of Samara started a campaign against Valery Grishko, a director at Samara academic theater, who played a bishop in Leviathan. The actor created a “cynical and dirty parody of Russian Orthodox bishops, was bullying toward the government and the Orthodox Church,” they said.