Cheryshev hammered in a fourth with the outside of his left in stoppage time and Golovin capped a magnificent performance by bending in a last-gasp free-kick.
It was the host nation's first World Cup win since 2002, and it raises the hope that Russian Federation can get past the group stage for the first time since the Soviet era.
Russian Federation looked more purposeful from the start as they poured into the vast spaces on both flanks and it was no surprise when they opened the scoring after 12 minutes as Gazinsky was left unmarked to nod home Golovin's deep left-wing cross.
The first goal arrived after 12 minutes from a neat header by unheralded 28-year-old Yuri Gazinsky, and came from a pinpoint cross from Golovin, the CSKA Moscow midfielder, who has been linked with a host of top English Premier League clubs. Golovin added the fifth from a free kick.
In the dying minutes of the match, Cheryshev struck his second before a stunner of a freekick from Aleksandr Golovin 25 yards away from the post to make it 5-0 for Russian Federation.
For the time in World Cup history, there will be Video assistant referees to help make officiating calls.
Dzagoev pulled up with an injury midway through the first half and was substituted in the 24th minute.More news: Manchester City to start title defence at Arsenal
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What's next? The other two sides in Group A - Egypt and Uruguay - open their campaigns when they meet at midnight tonight (NZT).
Saudi Arabia's route to the knockout round is much more hard.
Putin told the 80,000-strong crowd at the Luzhniki Stadium before the match kicked off: "I congratulate all of you at the start of the most important championship in the world".
Russian Federation opened their campaign in emphatic fashion, thrashing Saudi Arabia 5-0 in front of 78-thousand enraptured fans at Moscow's Luzhniki Arena.
- Saudi Arabia have not won any of their last 11 games at the World Cup (W0 D2 L9).
Egypt coach Hector Cuper said he could assure fans "almost 100 percent" Salah will play some part in Yekaterinburg.