Fabio Capello's side had to wait until the 82nd minute to break the deadlock as Alexander Kerzhakov came off the bench to head home the winner from close range.
Alan Dzagoev and Aleksandr Kokorin both wasted chances in the first half, while Slovakia had good spells of possession in both halves and saw Vladimir Weiss go close on two occasions to break the deadlock.
Russia continue warm-up for the finals in Brazil with another friendly against Norway on Saturday in Oslo, before facing Morocco in Moscow on June 6.
Vladimir Weiss and Miroslav Stoch both wasted early chances for Slovakia in the opening 15 minutes, with the former seeing his deflected shot in the seventh minute well saved by Igor Akinfeev.
Russia's first chance on target came in the 18th minute as Dzagoev saw a low effort easily saved by Jan Mucha, and Kokorin was also unable to stretch the goalkeeper nine minutes later.
The visitors worked their way back into the game in the closing stages of the half only for Akinfeev to deny Robert Mak and Stoch.
Russia had the ball in the net in the 57th minute as Kokorin fired low past Mucha from close range, but the 23-year-old's effort was disallowed after being flagged offside.
Oleg Shatov spurned three presentable opportunities at goal as the hosts became desperate but the breakthrough finally came with eight minutes to play.
Substitute Kerzhakov peeled away from his marker inside the penalty area to head Maksim Kanunnikov's left-wing cross into the bottom corner from five yards.
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