Hosts Russia were one of the surprises of the summer's World Cup, when they came close to booking a place in the last four before exiting the competition 4-3 on penalties to Croatia.
Stanislav Cherchesov's men were intent on taking up where they had left off and the visitors looked well on their way when Valencia winger Denis Cheryshev opened the scoring with a well-struck volley after just 13 minutes.
Serdar Aziz gave Turkey a lifeline when he levelled four minutes before the interval with a fine strike from just inside the area as Russia failed to clear.
But only eight minutes later slack defending saw Yuri Gazinski's short pass find Dzyuba unmarked on the right side, the captain finishing with calm to stun the hosts.
Turkey should have levelled in the dying seconds when Hakan Calhanoglu pounced on a fortuitous rebound inside the area but Russia 'keeper Andrei Lunev did well to get down quick and thwart his low snap shot.