Miroslav Stoch scored the only goal as Russia suffered a shock home defeat at Moscow's Lokomotiv Stadium.
The Fenerbahce midfielder took advantage of keeper Igor Akinfeev's mistake to find the net in the 27th minute and Russia failed to make the most of their dominance as Slovakia held firm, with keeper Jan Mucha to the fore.
The hosts started strongly in front of 30,000 fans, though a series of early corners came to nothing.
Mucha parried Roman Shirokov's fierce drive and was forced into action once again by Alan Dzagoev's 24th-minute effort.
Shirokov and Dzagoev combined to create the next chance for Igor Semshov, who fired wastefully over - and Slovakia then took the lead against the run of play.
The attack seemed to have broken down as Akinfeev claimed the ball, but the keeper's misplaced throw out was intercepted by Stoch, who advanced to the edge of the area and found the top corner of the net.
Pavel Pogrebniak shot wide from close range after a one-two with Konstantin Zyryanov, though Akinfeev was then forced to save well from Juraj Kucka.
Zyryanov had a goal correctly disallowed for offside before Yury Zhirkov shot wide when well placed.
Zhirkov's wayward shot then turned into a useful pass for Andrey Arshavin, but the Arsenal playmaker's shot on the turn flashed inches wide.
Vladimir Bystrov's effort was deflected wide with 20 minutes remaining, and Mucha had one more key contribution to make as he kept out Dzagoev's late free-kick.
The below-par Arshavin was brought down by Zdenko Strba just outside the penalty area, and Dzagoev's low effort was superbly saved by the impressive keeper to earn his side a second successive victory to kick off their Group B campaign in perfect style.