Yuri Gazinsky made the point with a header from a pass by Alexander Golovin.
No speech was delivered at the ceremony but Russian President Vladmir Putin gave a brief welcome before the match to the thousands of spectators, which also included Saudi Crown Prince Mohammed Bin Salman. Putin then reached over to shake the prince's hand. He later chipped the ball in for Russia's fourth goal in stoppage time.
The forward was soon replaced by Dzyuba, who struck nearly immediately by heading another Golovin cross beyond a helpless Muaiouf in the 71st minute.
The best player on the pitch, though, was central playmaker Aleksandr Golovin, who was rewarded for his offensive sparkle with a wonderful goal from a free kick just before the final whistle to cap off the scoring.
The first World Cup on Russian soil will be played in 12 stadiums located across 11 cities.
The completely refurbished Luzhniki Stadium in Moscow hosted the opening match, against the two lowest-ranked sides in the tournament.
But they will be aware that the Asian qualifiers look desperately ill-equipped to trouble Egypt or group favorites Uruguay, who meet tomorrow, and that there is still much to be done for the hosts to secure progress to the knockout stage. Saudi Arabia, the United Arab Emirates, Bahrain, Kuwait, Lebanon, Jordan, and Iraq all voted against the Moroccan bid. But that's not the case with Russian Federation, whose two remaining games are against group favorites Uruguay and Egypt.
Russia's players were significantly taller and heavier than the Saudis, a help when challenging for the ball and when the penalty area was crowded.
The home fans finally held their breath in the 56th when Taiseer Al Jassam just failed to slide into a sharp cross in front of an empty Russian goal on the only serious chance for the Saudis.
Russian Federation lost Dzagoev to a hamstring injury midway through the first half, the CSKA Moscow midfielder clutching his hamstring as he crumpled to the turf.
But if past history is anything to go by, Pizzi could well become the next coaching casualty of this tournament if Saudi authorities react with the emotive anger displayed in the past.