During the game, Putin's gestures, seemingly directed at Saudi Prince Mohammed bin Salman seated next to him, sparked a wave of online commentary.
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. "I already felt very happy when I knew I could come here with my squad, but I never, ever dreamed of something like this".
Many supporters were visiting Russian Federation for the first time. But high points for Russian athletes have been rare since the country's image was tarnished by doping, with numerous Olympic medals stripped for drug use.
The World Cup features 32 teams, including holders Germany, competing in 64 games over 32 days.
The Saudis, who last won a match at the World Cup in 1994, failed to get a single shot on target.
While the mood is dark in the Saudi Arabia camp, it could not be brighter for the hosts.More news: Shujaat Bukhari: Police releases photos of suspected attackers
A lack of an attacking threat in recent outings was a primary concern for Russian Federation going into a group campaign in which it will also face Egypt and Uruguay.
But his replacement, former Real Madrid winger Cheryshev, celebrated his recent recall to the national side with Russia's second goal two minutes before half-time.
"He played with a back four, which is normal against a team like Saudi".
Juan Antonio Pizzi, Saudi Arabia's coach, described a "feeling of shame". Artem Dzuyba netted a third before terrific late efforts from Cheryshev and Aleksandr Golovin wrapped up Russia's first win since October to launch their Group A campaign in style.