Denis Cheryshev, who came on as a substitute in the 24th minute, scored two great goals, including an absolutely handsome curling effort in stoppage time at the end of the second that is likely to be one of the goals of the tournament.
"It is left to us to be positive and start thinking about the next match", he said.
Dzagoev's replacement, Cheryshev, certainly stepped into the breach by scoring two goals after replacing the CSKA Moscow midfielder, including a stunner to make it 4-0, but the hosts can not afford to have their resources depleted even further by the loss of one of their best players.
The host nation have broken Russian and Soviet records by going winless in seven matches and dropping to 70th in the world, meaning no World Cup coach faces as much pressure as Stanislav Cherchesov.
Warm-up losses to Italy, Germany and Peru have not made for the best preparations for the Saudis as they chase a first victory in the World Cup finals since 1994 in their first appearance in the showpiece event since 2006.
Those fears were soon washed away, at least for the time being, as Alexander Samedov and Fedor Smolov carved out early chances only to be denied by desperate sliding blocks.
Putin smiled several times as he watched the game in a VIP box with Saudi Crown Prince Mohammed bin Salman. The cross that set up the tournament's opening goal came from CSKA Moscow's 22-year-old Aleksandr Golovin.
Melting Ice Caps: Coastal cities DOOMED as Antarctica thaws RAPIDLY
Altogether, across the continent, 34,000 square kilometers (13,000 square miles) of the ice shelf has been lost since the 1950s. Pippa Whitehouse, a member of the IMBIE team from Durham University, according to a University of Leeds press release.
Possible tornado rips through Wilkes-Barre; multiple buildings damaged
A powerful storm damaged the Panera Bread restaurant in the Arena Hub shopping area in Wilkes-Barre Township, Pa., on Wednesday. Township Police Chief Ron Smith said the six injured people suffered only minor wounds and were not hospitalized.
Ronaldo: Cristiano is unique but Messi provides more magic
I want Argentina to win the title because this his last chance for the little genius Messi to win a major trophy for his country. Messi is Barcelona's all-time leading goal-scorer and Ronaldo holds the same status for Real Madrid .
Saudi Arabia's midfielder Taisir Al-Jassim (L) vies with Russia's midfielder Aleksandr Golovin during the Russian Federation 2018 World Cup Group A football match between Russian Federation and Saudi Arabia at the Luzhniki Stadium in Moscow on June 14, 2018.
Dzagoev pulled up with an injury midway through the first half and was substituted in the 24th minute.
But his replacement, former Real Madrid winger Cheryshev, celebrated his recent recall to the national side with Russia's second goal two minutes before half-time.
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.
The crowd were then treated to a fabulous finale as Cheryshev cleverly steered home the fourth with the outside of his foot before Golovin capped a great personal performance by curling his free kick beyond the wall to get the month-long Moscow party well and truly started.