Artyom Dzyuba made an instant impact off the bench to make it 3-0 in the 71st with a header off Golovin's cross. Substitute Denis Cheryshev scored twice - the first after a piece of skill that would have graced a Lionel Messi highlights reel - and the second a superb shot in stoppage time.
After an opening ceremony which included British pop icon Robbie Williams, it took just 12 minutes for the tournament's first goal, with Gazinsky opening his worldwide account with a well-taken header across goal.
The 2018 FIFA World Cup kicks off Thursday in Russian Federation, with a field of 32 teams from around the globe competing in a monthlong tournament that concludes July 15.
Both will also be hoping for victory at the Luzhniki to end a barren run at the global finals: Russian Federation have not won at the World Cup since 2002, while Saudi Arabia's last global finals' victory came at US 1994. Golovin added the fifth from a free kick.
Canada will host men's World Cup matches for the first time, while Mexico gets its first taste of the event since 1986.
Prosecutor Says No Evidence of Foul Play in Anthony Bourdain's Death
I didn't talk to him this week but all I know was he was so happy last week. "But there's been nothing yet", he said. Argento had accused Hollywood film producer Harvey Weinstein of sexual assault and become a vocal critic of his.
In a departure from what has become the norm for opening ceremonies, this year's inauguration was shorter and less flamboyant than what many would have expected, lasting some 30 minutes and featuring British pop star Robbie Williams as the celebration's highlight.
In a World Cup first, the video assistant referee system was available for the match between the tournament's lowest-ranked teams, though it wasn't used. Preparation for the World Cup will not require major construction work because the 16 stadiums where matches will be played already exist.
But with the five goals against the Saudis, a draw could be enough against either of its two remaining opponents.
Russian Federation can now prepare to face Egypt next week full of confidence while Saudi Arabia need to find instant improvement if they are to avoid further embarrassment against Uruguay.
The other Group A teams Uruguay and Egypt meet Friday in Yekaterinburg.
Saudi Arabia will take the field on Thursday as lengthy +800 underdogs.