Russia needed just a point to seal their qualification and Cheryshev opened the scoring in the ninth minute to set them on their way, before he set up Magomed Ozdoev for the second at GSP Stadium on Sunday.
Any slim hopes of a Cyprus fightback were extinguished in the 27th minute when Kostas Laifis was sent off for a poor foul on Sergey Petrov.
The Russia full-back had to be substituted and team's official Twitter account posted a photograph showing the wound in his leg.
Тем временем на скамейке запасных сборной России pic.twitter.com/i97TNUAltF— Сборная России (@TeamRussia) October 13, 2019
Petrov and his team-mates can look forward to next year's finals after Russia guaranteed they will finish in the top two of Group I as Artem Dzyuba, Aleksandr Golovin and Cheryshev added late goals.
Belgium, group leaders with maximum points, had already secured their place at the tournament with a 9-0 defeat of San Marino on Thursday.