"We qualified from the group, thus, achieving the minimum aim ahead of time. We played two games at a very high level, with a good result," Gazzaev said.
He added that Russia could now face Spain or Portugal in the knockout stage.
"This is going to be some serious football," Gazzaev said.
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Russia beat Egypt 3-1 in St. Petersburg on Tuesday and are now top of Group A with six points from two games and are almost certain to reach the round of 16 if Uruguay avoids defeat against Saudi Arabia on Wednesday. Denis Chernyshev and Artem Dzyuba each scored for the team, while Egypt's defender Ahmed Fathy netted an own goal.
The 2018 FIFA World Cup kicked off on June 14 and will continue until July 15, with the matches set to be played in 11 cities across Russia. Hosts Russia will now take on Uruguay at the 42,000-seat Samara Arena on June 25.