0250 GMT August 18, 2018
Substitute Denis Cheryshev scored two brilliant goals after Yury Gazinsky had headed the World Cup's first after 12 minutes, Reuters reported.
Artem Dzyuba was also on target a minute after coming on as a 70th-minute substitute and Aleksandr Golovin completed a memorable day by smashing in a free kick with the last action of the game.
The result equaled the best in a World Cup opening game - Brazil beat Mexico 5-0 in Geneva in 1954 - and watching Russian President Vladimir Putin wasted no time in phoning coach Stanislav Cherchesov to congratulate him.
"We are not getting ahead of ourselves. 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.
Constantly criticized in the ragged build-up games, Cherchesov had promised improvement once the phoney war was over and his energetic players duly delivered.
"We were tighter, more disciplined, more mature," he said. "To be good is one thing, to be good at the right time in the right place is another."
It also helps if you are facing a team who have now lost eight and drawn two of their last 10 World Cup matches, whose defending fell painfully short of World Cup standard and whose attack was non-existent.
Coach Juan Antonio Pizzi does not speak any Arabic but it will not take long for them to become aware of his Spanish description of their performance as shameful.
Russia's fans, forced to suffer years of poor performances in friendlies that left their team an embarrassing 70th in the world rankings will not care about the quality of the opponents.
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.
Russia 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.