Former President Barack Obama once dismissed post-Soviet Russia as a mere “regional power,” but that isn’t how things look today. Vladimir Putin has gone global in recent years, launching a Russian-style charm offensive in far-flung locales where the Kremlin’s influence had been all but written off. Russian voices, fingerprints and footsteps have been showing up over much of the Middle East and Europe, parts of Africa and even in Latin America.

Moscow has found numerous openings and is busily exploiting divisions within the...