Slutskiy, who took over from Fabio Capello in August, looked on course to maintain his unbeaten record when Fedor Smolov put the hosts ahead in the first half.
Yet a fine second half performance produced goals for Croatia's Nikola Kalinic, Marcelo Brozovic and Mario Mandzukic to maintain coach Cacic's 100 percent record after three games since he took over in September.
Slutskiy made numerous changes to the side that beat Portugal 1-0 in Krasnodar on Saturday, and despite making a nervy start, Russia opened the scoring with a lovely team goal after 14 minutes.
Roman Shirokov's intelligent pass found Alexei Ionov galloping down the right. His cross was perfectly weighted for the on-rushing Smolov, who planted a header past Croatia keeper Lovre Kalinic for his fourth international goal.
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Midway through the first half, Juventus striker Mandzukic had a great chance after being put through one-on-one with Yuri Lodygin but the Russian goalkeeper produced a fine save.
Croatia equalised 12 minutes after halftime as Russia were caught on the counter-attack. Sime Vrsaljko found himself clear down the right flank and crossed for Kalinic, who turned the ball past substitute goalkeeper Artem Rebrov from close range.
Three minutes later, the visitors took the lead after a stunning goal by midfielder Brozovic, who collected the ball on the edge of the penalty area, before curling a fine effort past Rebrov.
Mandzukic sealed victory for the Croats in the 82nd minute. Again Cacic's side broke down the right and, from close range, the 29-year-old scored his 20th goal for his country.
(Reporting By Dmitriy Rogovitskiy, editing by Ian Chadband)
