Kokorin gave Russia the lead after six minutes with a long-distance strike which deceived Czech goalkeeper Petr Cech.
The Czechs equalised soon after halftime when Necid laid the ball into the path of Rosicky and the 35-year-old produced a fine right-foot finish to beat Russia's substitute keeper Guilherme.
Ukrainian-born Roman Neustadter was brought on to make his debut for the hosts after receiving his Russian passport on Monday, partnering Sergei Ignashevich in the centre of defence.
But he was at fault for the Czech winner in the final minute.
Necid nipped in front of the Schalke 04 player and deflected the ball past Guilherme from close range to give his team a deserved victory.
Russia play their final Euro 2016 warm-up match against Serbia on June 5 in Monaco.
Their first game in European Championship Group B is against England on June 11 in Marseille. The Czechs start their Group D campaign against holders Spain on June 13.
(reporting by Dmitriy Rogovitskiy, Editing by Ed Osmond)
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