Yuri Gazinskiy and Cheryshev struck first-half goals to open their international accounts at Luzhniki Stadium, Artem Dzyuba adding a third 89 seconds after his introduction as a second-half replacement, but an injury to star man Alan Dzagoev will cause concern for Russia.
Dzagoev hurt his hamstring in the opening period and could be a doubt for the rest of the tournament, depending on the results of a scan.
Gazinskiy headed in Golovin's cross to score the first goal of the 2018 event, with Dzagoev's replacement Cheryshev lashing in the second before the interval.
Russia started the World Cup as the lowest-ranked team in the tournament but the hosts were a class above Saudi Arabia, Dzyuba's header stretching the lead in the 71st minute.
Cheryshev showed he could be the man to fill Dzagoev's shoes with a fantastic injury-time goal, finding the top corner with a sublime shot with the outside of his left boot, before Golovin bent in a brilliant free-kick to finish the match in style.
Juan Antonio Pizzi's men struggled to create a chance of note, failed to record a shot on target and gave away possession on countless occasions, enabling Russia to record their first World Cup win since 2002.
Russia took the lead in the 12th minute with the first shot on target of the World Cup.
Golovin kept a cleared corner alive and delivered a floated cross to the back post, with Gazinskiy exploiting slack Saudi marking to plant a firm header past Abdullah Al-Mayoof.
But there was a major blow for the hosts mid-way through the half as star playmaker Dzagoev limped off with a hamstring injury, to be replaced by Cheryshev.
Russia had a penalty appeal turned down in the 35th minute, but Saudi captain Osama Hawsawi got enough on the ball when he challenged Golovin to avoid censure from VAR.
Stanislav Cherchesov's side did not let Dzagoev's injury affect them and they scored the second goal their enterprising play deserved three minutes before the break.
After a fine team move, Roman Zobnin slipped the ball to former Real Madrid winger Cheryshev, who evaded a pair of poor challenges in the box before slamming an unstoppable finish into the roof of the net.
Salman Al Faraj headed a decent chance over the crossbar as Saudi Arabia sought a reply after the interval.
But Russia's win was all but sealed in the 71st minute with Golovin again the creator, the CSKA Moscow midfielder producing a lovely centre from the right that allowed Dzyuba to nod in within two minutes of his introduction.
Cheryshev was a threat throughout and he hit an early contender for goal of the tournament with a brilliant strike that gave helpless Al-Mayoof no chance.
And the Saudi goalkeeper was beaten again in the fourth minute of stoppage-time, Golovin adding to his two assists with a curling free-kick that left brought the home fans to their feet once again.
Key Opta stats:
- 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).
- Russia 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 have not won any of their last 11 games at the World Cup (W0 D2 L9). Their last win in the competition dates back to the 1994 group stages.
- Russia's Denis Cheryshev became the first substitute to score a goal in the opening match of a World Cup tournament.
- Artem Dzyuba scored for Russia just 89 seconds after coming off the bench; the quickest goal by a substitute at a World Cup since Marcin Zewlakow for Poland v USA in 2002.
- In this game, Sergei Ignashevich became the oldest player to ever appear in a World Cup match for Russia/USSR (38 years & 335 days old).