Midfielder Dzagoev's strike from 25 metres in the 71st minute was enough to eliminate Slovakia from the competition and avenged the Russians' 1-0 home loss to the same opponents at the start of qualifying.
The jeering crowd in the northern Slovak town of Zilina witnessed fast-paced end-to-end play with numerous long-range volleys struck on both sides that had earlier failed to produce a goal.
The Slovaks went closest to scoring at the end of the first half after Fenerbahçe's Miroslav Stoch took a pass within the box, but wasted the chance to end Slovakia's three-match goal drought.
Zenit St Petersburg midfielder Roman Shirokov missed a chance for Russia with 25 minutes left after a fizzing pass from the left by Dzagoev.
Russia, who lead the group on 20 points, two above Ireland, play their last group match in Moscow on October 11.
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