Cheryshev scored a stunning goal with the outside of his left foot in time added on and Golovin sealed the rout with a goal direct from a free-kick.
Luzhniki exploded into raptures after only 12 minutes when Yury Gazinsky headed home for the opener.
Stat chat Cheryshev became the first substitute to score in an opening World Cup game, while team-mate Artem Dzyuba did the same, heading home Russia's third just 90 seconds after coming on in the second half.
In the dying minutes of the match, Cheryshev struck his second before a stunner of a freekick from Aleksandr Golovin 25 yards away from the post to make it 5-0 for Russian Federation.
Russian Federation lost Dzagoev to a hamstring injury midway through the first half, the CSKA Moscow midfielder clutching his hamstring as he crumpled to the turf.
But they will be aware that the Asian qualifiers look desperately ill-equipped to trouble Egypt or group favourites Uruguay, who meet tomorrow, and that there is still much to be done for the hosts to secure progress to the knockout stage.
The tournament's 64 games will be played at 12 stadiums throughout Russian Federation, with the opening match and the final held at Moscow's Luzhniki Stadium.
Cities such as Saransk were sleepy outposts with decaying buildings until the World Cup reconstruction put them firmly in the 21st century.
- Russia's victory means that the host nation has never lost their opening match in any of the 21 editions of the World Cup (W16 D6 L0).
- Russian Federation scored more than twice the amount of goals in this game (5) than they did in the entire 2014 World Cup finals, over three games.
Saudi Arabia also play Uruguay and Egypt in Group A.
Saudi Arabia captain Osama Hawsawi was on his guard against Golovin - and Argentine referee Nestor Pitana didn't bother to consult the video assistant despite Golovin's penalty appeal.
Brazil and their superstar Neymar are seeking a sixth global crown while Germany, who won their fourth World Cup in Brazil four years ago, will be determined to draw level with the Brazilians.
Coach Juan Antonio Pizzi had earlier said that Saudi Arabia would play without fear on Thursday.