New Zealand almost pulled a goal back in the 77th minute but Akinfeev produced a fine save to keep out Ryan Thomas' effort, before Tommy Smith saw his header cleared off the line a minute later.
The hosts now take on Portugal on Wednesday, with Fernando Santos' men taking on Mexico in their Group A opener on Sunday.
New Zealand did force Igor Akinfeev into a couple of relatively easy early saves, but Russian Federation took control of the match in the 31st minute.
The hosts won the opening match of Group A at the Krestovsky Stadium in front of Russian President Vladimir Putin in Saint Petersburg.
Russian Federation took the lead through Michael Boxall's first-half own goal.More news: Kohli, Sharma collapse in high-octane Champions Trophy final
Smolov was named Budweiser Man of the Match, while New Zealand now have two unwanted Tournament of Champions records to their name, having gone 513 minutes without scoring in the competition and ten matches without winning.
But weak defending allowed Russia's Fedor Smolov to slip past the New Zealand defence and tap in a low cross by team mate Alexander Samedov in the 69th minute.
According to Infantino, the stadiums in Russian Federation produce a big impression and both tournaments offer a good chance for all to give a new impetus to football development.
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.
"I believe we will have the same team as we did in this game against Portugal".More news: Microsoft's Albert Penello on Xbox One's "post-X" future
New Zealand had their first major chances when Ryan Thomas' powerful effort forced an impressive save from Igor Akinfeev and Smith's header from the resulting corner was hacked off the line by Yuri Zhirkov.
After this match, New Zealand are now without a win in 10 Confed Cup matche.
New Zealand nearly clawed themselves back in the game twice - first Ryan Thomas had a brilliant effort saved by Igor Akinfeev and then Tommy Smith had a powerful header cleared off the line by Yuri Zhirkov. We played the match we have been planning for.
"We don't get a lot of opportunities to play against top teams, so this has been an unbelievable experience for us, and we're going to learn a lot from this game".
"The fact the leader of the country came to the match is on the one hand an additional positive, but it is also an additional responsibility", Russian Federation manager Stanislav Cherchesov said.More news: Champions Trophy : India bowl against Pakistan in final
Russia's victory showed it can win a big game at home, even if Putin is in the arena.
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