The score of 3-0 seemed to remain flashing unchanged until the final whistle in the game, but in the three-minute stretch of the added time Russia's Denis Cheryshev upped the score to 4-0 with a swift shot of the ball over the Saudi Arabian goalkeeper.
Cheryshev and Golovin hit a further two in injury time, the latter from a free kick, to complete the visitors' misery.
The other Group A teams Uruguay and Egypt meet Friday in Yekaterinburg.
Taking this into account, and the fact that Russian Federation are expected to come away with the win, Cheryshev looks a good bet at 7/2 to score anytime in this match.
Russian Federation took the lead in the 12th minute with the first shot on target of the World Cup.
Soon after Russian Federation took the lead as a superb cross from the left from Aleksandar Golovin found Gazinskiy at the back post who headed home his first worldwide goal.More news: L.A. District Attorney’s Office Reviewing Sex Crimes Case Against Sylvester Stallone
Minutes before the game started in the Luzhniki Stadium in Moscow, Putin said: "I congratulate all of you at the start of the most important championship in the world".
12 minutes into the game, Yury struck his first-ever global goal and also the first in the 21st edition of the quadrennial worldwide football tournament. "We saw that we are on the right track but we should forget this game and concentrate on the next one", Cherchesov said after ending a seven-game winless run in emphatic style.
Even Russian President Vladimir Putin, sitting next to Federation Internationale de Football Association president Gianni Infantino and Crown Prince of Saudi Arabia Mohammad Bin Salman Al Saud, found himself looking over like "hey, what can you do?"
Both sides tried to assert their dominance in the opening exchanges by holding onto the ball and dictating the tempo of the game.
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.
While Western countries chose not to send senior representatives to the opening ceremony, there are no sporting boycotts like the one that marred the 1980 Moscow Olympics, or doping bans of the kind that excluded many Russians from the last two Olympic Games.
That's right; while you would assume his World Cup duties with France are first and foremost in his mind, the Atletico Madrid striker made a big deal of releasing a big-budget, over-produced trailer about his club allegiances.