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'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.
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.
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.
