At 14:04 of the first period, Grigorenko connected with former Quebec Remparts teammate Nikita Kucherov, criss-crossing while entering the zone and taking a pass from the winger before dishing it back door on the give-and-go for the 1-0 lead.
The 6-foot-2 centerman continued to work the middle of the ice responsibly as the Slovaks fought their way back to even the game 2-2 with a strike in the final minute of regulation.
After Slovakia went shorthanded in overtime, Grigorenko came over the boards to take a key offensive zone draw with 14.5 seconds to play. The talented pivot came through, winning the puck back to the point and drifting through the low slot as a Albert Yarullin blast found the back of the net allowing Russia to escape with the two points.
Grigorenko finished the game with team-bests five shots on goal and a 69% success rate at the faceoff circle (11/16).
Sabres goaltending prospect Andrey Makarov served as backup while Andrei Vasilevskimade 32 stops to earn Russia's Player of the Game honors.
Russia returns to action Friday versus Team USA, who are captained by fellow Sabres prospect Jake McCabe.