In an attempt to clear the ball from the six-yard box, Skriniar sloppily deflected a dangerous cross by Arsen Zakharyan past goalkeeper Marek Rodak, handing Russia the lead in the 24th minute.
Slovakia dominated possession, their best chances coming late in the first half. Russian goalkeeper Matvei Safonov denied Lukas Haraslin with a diving save, and a low strike by Slovak midfielder Ondrej Duda sailed just wide of the far post.
Slovakia pressed in the second half for an equalizer, but they failed to capitalise against a revitalised Russian side under the leadership of new coach, Valery Karpin.
With their win on Friday, Russia remain second in Europe's Group H with 16 points, trailing Croatia on goal difference. Slovakia are in fourth place with nine points.
In the next round of games on Monday, Russia face Slovenia while Slovakia take on group leaders Croatia.