The game was scored with an own goal by New Zealand's Michael Boxall at the 31st minute of the game.
Days earlier, Putin had called for better results from the world's 63rd-ranked team to impress the Russian public.
"We've worked hard and I believe now we're well-prepared for the opening game", Zenit midfielder Igor Smolnikov told Russian TV. "We could have been more aggressive".
The other two Group A teams include European champions Portugal spearheaded by the global superstar Cristiano Ronaldo and the most decorated CONCACAF team Mexico.
In addition to the opening game, St. Petersburg will host two other matches during the group stage of the tournament: Cameroon vs Australia on June 22, and New Zealand vs Portugal on June 24.
New Zealand are now without a win in 10 Confederations Cup matches and struggled to make an attacking impression.
After New Zealand lost possession in a risky area, Poloz's neat touch released Glushakov, who clipped his finish over Marinovic and off the post before Boxall bundled the ball into his own net as he and Smith frantically tried to clear. A three-player race to meet the rebound off a post saw the sliding Boxall's trailing right arm get the final touch in a bundle of bodies.
Playing in the tournament for the first time, Russian Federation were on the front foot and had two early chances cleared off the line in the opening 10 minutes.
First, Aleksandr Samedov's corner found Viktor Vasin, whose diving header from 12 yards struck the inside the post before being hacked clear off the line by Michael McGlinchey. Tommy Smith then tidied up when Poloz poked a close-range shot slowly past Marinovic. In New Zealand, the match will be streamed on Sky Go NZ, and while the match isn't being shown online in Russian Federation, you can find viewing options for other countries by checking out LiveSoccerTV.
On Saturday, Russia marched on toward hosting football's biggest party when the rehearsal tournament, the Confederations Cup, kicked off in the $750 million new stadium in Putin's native St. Petersburg.