Cheryshev held the ball as two covering Saudi defenders slid by in the 43rd minute and shot past goalkeeper Abdullah Al Muaiouf's right side. He later curled in a shot with the outside of his foot for Russia's fourth goal in the 91st minute.
After Cheryshev's first goal, Artyom Dzyuba made an instant impact off the bench to make it 3-0 in the 71st with a header from Golovin's cross. Golovin added the fifth from a superb 94th-minute free kick he curled around the wall and inside the near post.
Following a lavish opening ceremony featuring British singer Robbie Williams, Putin welcomed visiting fans and promised Russia would be a "hospitable and friendly" host.
In a World Cup first, the video assistant referee system was available for the match between the tournament's lowest-ranked teams, though it wasn't used.
The Saudis, who last won a match at the World Cup in 1994, failed to get a single shot on target.
The win for Russia is a big boost to their hopes of advancing. They face Egypt on June 19 and Uruguay six days later. But with the five goals against the Saudis, a draw could be enough against either of their two remaining opponents.
Saudi Arabia's route to the knockout round is much more difficult. The Saudis face Uruguay on June 20 and Egypt on June 25. Egypt and Uruguay played overnight in Yekaterinburg.
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.
That had an effect on Gazinsky's goal, which came after the Saudis struggled to clear a Russian corner, and the header from the tall and stocky Dzyuba.
One concern for Russia, however, was the loss of creative midfielder Alan Dzagoev, who limped off in the first half with a hamstring injury.
Saudi Arabia sent some players to the Spanish league for experience this season but they didn't get much playing time while there.
- AP
