The Russian team is playing its next match against Portugal at Spartak Arena in Moscow, while New Zealand is going to the Russian southern resort of Sochi, where they are scheduled to play Mexico at the Fisht Stadium.
"First of all, let me express gratitude to [FIFA President] Gianni Infantino for being committed to the sport ideals and for helping in the preparations of this landmark event", Putin said at the opening ceremony of the cup.
The Russian president made it a national priority to win hosting rights for the 2018 World Cup in a fierce Federation Internationale de Football Association bidding contest seven years ago.
Meanwhile, New Zealand are appearing for the fourth time, although the All Whites are still hunting for a first win in the eight-team competition.
New Zealand's Michael Boxall got a touch on the ball and was credited with an own goal.
Putin said this week he expected Russian Federation to play like warriors and that fans expected better results from the world's No. 63-ranked team.
Very good Chile side are priced at 9/4 after claiming back to back Copa America trophies, while Mexico and hosts Russian Federation are both priced at 1/1.
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Russian Federation deserved its first-half lead after twice having shots stopped on the line in the opening 10 minutes.
Group A rivals Portugal and Mexico meet in Kazan on Sunday.
Cherchesov's men were too strong for a limited New Zealand side and were unlucky not to take an early lead, Viktor Vasin heading against the post from an Alexander Samedov corner, with Michael McGlinchey clearing off the line.
Russian Federation came so close again when Dmitry Poloz chested down a cross from right and stabbed his effort from six yards past Marinovic, only for Tommy Smith to get back on the line to clear. The president prefers ice hockey and saw a team that carried fervent hopes of an Olympic title lose on home ice in the quarterfinals at the 2014 Sochi Olympics.