Tune in to Jason Pine weekday mornings from 9 on Radio Sport, for all the latest from the FIFA World Cup.
It was Russia's first World Cup win since 2002 and ended a seven-game winless run in all matches.
Yuri Gazinsky initially put Russian Federation ahead with a header from Alexander Golovin's cross when the Saudis failed to clear a corner in the 12th minute. After one goal scored by Russian Federation, the two men looked at each other, shrugged and then shook hands.
Russian Federation made it 3-0 in the 71st minute when Golovin chipped in a cross for giant striker Dzyuba to rise and head in from close range. Golovin added the fifth from a free kick.
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.
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Saudi Arabia not only voted for United 2026, but also campaigned against Morocco in the vote for the host of the 2026 World Cup. The visitors failed to get a single shot on taget in the contest between the two lowest-ranked teams at the tournament.
But with the five goals against the Saudis, a draw could be enough against either of its two remaining opponents.
Egypt, the other team in Group A, face the South American side on Friday in Ekaterinburg, and Egypt and Saudi Arabia then play each other in a "Red Sea derby" match in Volgograd on 25 June.
A lack of an attacking threat in recent outings was a primary concern for Russian Federation going into a group campaign in which they will also face Egypt and Uruguay. Aware that Saudi Arabia's defenders were unlikely to challenge him, the attacker curled in a attractive shot from the edge of the box with the outside of his left foot to put his team up four.
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. Flag-waving supporters from all over the world let out huge cheers and chanted "Russia!" as the host team scored twice in the first half at the city's Luzhniki Stadium.
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.