Just 12 minutes into the match, the Saudi backline got caught out of place after a corner kick, leaving a wide open opportunity for Yury Gazinsky to score the opening goal of the tournament.
Dzagoev hurt his hamstring in the opening period and could be a doubt for the rest of the tournament, depending on the results of a scan.
It was Gazinsky's first goal of his global career.
Artyom Dzyuba made an instant impact off the bench to make it 3-0 in the 71st with a header off Golovin's cross.
Russia, he explained, have been preparing for their moment of history in their home World Cup.
"We won our first game at [last summer's] Confederations Cup and it didn't take us too far". Gazinsky put home a header, and the Saudi Arabian goalie simply couldn't get to it despite a good effort.
One concern for Russian Federation is that creative midfielder Alan Dzagoev picked up a suspected hamstring injury, but that was what allowed Cheryshev to come off the bench and change the game.
It's hard to imagine Russian Federation beating the vastly superior Uruguayans, but if they can get a draw with Egypt, whose best player, Mo Salah, is fresh off a serious shoulder injury suffered in the Champions League final, they could scramble through to the next round.
And that strike sent fans wild over the former Real Madrid man. Aware that Saudi Arabia's defenders were unlikely to challenge him, the attacker curled in a lovely shot from the edge of the box with the outside of his left foot to put his team up four.
The heavy defeat leaves Saudi Arabia likely needing to beat Uruguay in their next match on Wednesday to stand a realistic chance of advancing beyond the group stage of the tournament. After one goal scored by Russian Federation, the two men looked at each other, shrugged and then shook hands.
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.
They inexplicably sacked Bert van Marwijk despite a successful campaign and then his successor Edgardo Bauza before the World Cup draw in November.