After New Zealand lost possession in their own half, Poloz's neat touch set up Denis Glushakov, who dinked his effort over goalkeeper Stefan Marinovic and off the post before Boxall bundled the ball into his own net.
The new football arena in St. Petersburg is hosting the opening and final matches of the Confederations Cup, also dubbed as the Tournament of Champions.
The game was scored with an own goal by New Zealand's Michael Boxall at the 31st minute of the game.
Federation Internationale de Football Association general secretary Fatma Samoura said on Friday that the estimated attendance for the tournament would be 65 percent, and that Russia's match against European champions Portugal had been sold out for weeks.
"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, whose team face Portugal on Wednesday, told reporters.
"Here, on modern football arenas there will be tough, honest, fair fight until the last minutes of the match", Putin said in Russian, adding that the eight-team, two-week event would "unite nations and continents, to promote the values of fair and handsome play".
Russian Federation made sure of victory in the second half, with Fyodor Smolov's third goal in four internationals ensuring president Vladimir Putin, who called for improved performances in the event, was applauding happily in the stands.
The Confederations Cup precedes the World Cup by a year to guarantee that the facilities and provisions of the host country are ready.
Champions Trophy: Australia vs Bangladesh, match abandoned due to rain
In his next over, Starc bowled Mehedi Hasan for 14 after Mustafizur Rahman had turned his back on an easy second the ball before. The frequent loss of partners at the other end was what hampered Bangladesh , as Starc went about cleaning up the tail.
Russian President Vladimir Putin and FIFA President Gianni Infantino officially opened the tournament.
"We knew the scenario with who was coming to the game", New Zealand coach Anthony Hudson said.
"Cristiano is totally focused in helping our team".
In the only match played Saturday, Smolov took his chance to shine.
Smolov had an effort ruled out for offside before the opening goal arrived just after the half-hour mark.
It helped the Russian team to seize the lead after twice having shots stopped on the line.
New Zealand is now without a win in four trips to the Confederations Cup, and next plays Mexico on Wednesday in Sochi.
Russia's victory showed it can win a big game at home, even if Putin is in the arena.
World Cup holders Germany and Australia complete the group.
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