Turkey coach Mircea Lucescu fielded a revamped young side who initially failed to impose themselves on the visitors despite a rowdy crowd in the Black Sea province of Trabzon.
FIFA World Cup quarter-finalists Russia took the lead in the 13th minute when Cheryshev rocketed a left-foot volley past goalkeeper Serkan Kirintili, silencing the home fans.
Centre-back Serdar Aziz equalised just before half-time when he unleashed a right-foot shot into the Russia net, but the visitors regained the lead shortly after the break.
A defensive misunderstanding left Dzyuba one-on-one with Kirintili and the striker fired the ball past the keeper to give the Russia a 2-1 advantage.
Despite controlling the majority of the game and making several chances late on, Turkey could not capitalise.
The Nations League, brainchild of UEFA, aims to boost interest in internationals, which are seen by clubs and fans as disruptive to the domestic football season.
To be held every two years, it features all 55 of Europe's national teams divided into four divisions split into four groups, with games to be played between September and November.
The four group winners will be promoted to the higher division, while the bottom four teams will be relegated to the lower division.
Sweden are the other side in League B Group 2, and they will host Turkey in their opener on Monday evening (Tuesday morning AEST).
