Russia on brink of finals with win in Slovakia
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.