Stoch capitalised on a bad mistake by Russia keeper Igor Akinfeev to fire the visitors ahead midway through the first half and against the run of play.
The usually reliable Akinfeev gave the ball away just outside his penalty box and an unmarked Stoch calmly blasted into the top corner.
The Russians dominated against the defensive-minded Slovaks for much of the game but wasted numerous chances to salvage at least a point.
"This is a fantastic result for us as Russia are one of the best teams in Europe," beaming Slovakia coach Vladimir Weiss told a news conference. "We have a young team but we're getting better with every day."
The win in Moscow gave Slovakia a perfect start in their qualifying campaign following a last-gasp 1-0 victory over Macedonia in Group B on Friday. Russia, who scraped past lowly Andorra 2-0 in their opener, now face a tricky trip to Ireland next month.
The home team began confidently, testing Slovakia keeper Jan Mucha several times from long range. The Slovaks were content to soak up the pressure and threaten on the counter-attack - their plan working to perfection with Stoch's strike.
The Russians increased the tempo after the break but poor finishing cost them dearly.
Russia captain Andrei Arshavin had the best chance for an equaliser late in the match but his low shot just missed the far corner. Konstantin Zyryanov also had a goal disallowed for a marginal offside by Belgian referee Frank de Bleeckere.
Russia coach Dick Advocaat was left to rue missed opportunities. "Obviously, I'm very disappointed," said the Dutchman, who succeeded compatriot Guus Hiddink in May after Russia failed to qualify for this year's World Cup.
"We had so many chances and couldn't score. Our opponents didn't create any of their own - we gave them their only chance and they took full advantage of it," he added.
"The luck was clearly not on our side tonight."
Russia also lost valuable points against Slovakia at the start of their 2006 World Cup qualifying campaign when they were held to a 1-1 draw in Moscow that eventually cost them a place in the finals in Germany.
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