Yuri Gazinsky initially put Russian Federation ahead with a header from Alexander Golovin's cross in the 12th minute.
Russian Federation welcomed the world to the World Cup today as cheerful fans streamed into Moscow and 10 other cities and President Vladimir Putin's government derided Western efforts to isolate him.
The tournament also offers Putin a chance to project Russian Federation as a global player that is accepted and respected even while being at odds with the United States.
One concern for Russian Federation, however, is that creative midfielder Alan Dzagoev appears to have injured a hamstring in the first half.
Dzagoev appeared to pull a hamstring off the ball and could be a doubt for the rest of the tournament.
By heading home just 89 seconds after entering the game, Dzyuba became the fastest substitute to score at a World Cup since 2002.
Saudi Arabia's next Group A encounter is against Uruguay in Rostov on June 20.
Cheryshev added a fourth with a superb finish before Golovin curled in a superb free-kick in injury time as President Vladimir Putin watched on in delight.
Saudi briefly threatened early in the second half when an inviting cross in from the right skipped just beyond the reach of Taiseer Al-Jassam. FIFA president Gianni Infantino addressed the crowd in Russian and said "football will conquer the entire world" during the cup.
Collecting a pass from Roman Zobnin on the left of the box he showed great composure to nonchalantly flick the ball inches above two prone defenders desperately sliding in, before lashing into the roof of the net.
Artem Dzyuba jumped off the bench to nod home to make it 3-0, then two fine strikes late on from Cheryshev and then Aleksandar Golovin, who also grabbed two assists, completed the rout.
Saudi Arabia recently lifted a ban on women driving, although that has been accompanied by a crackdown on dissent.
The forward was soon replaced by Dzyuba, who struck nearly immediately by heading another Golovin cross beyond a helpless Muaiouf in the 71st minute.
Minutes before Saudi Arabia kicked off the tournament against host nation Russia, Putin said: "I congratulate all of you at the start of the most important championship in the world".
'Football brings us altogether in one team, we are united by this spectacular game, ' he added.
Peter Tatchell was detained near Red Square on June 14 for holding a one-man protest to draw attention to what he said were human rights abuses committed against gay men in Russia's North Caucasus region of Chechnya.
Tatchell, famed for his pro-LGBT protests across the world, has been charged with breaking the law prohibiting protests during the World Cup.