The Manchester United attacking midfielder scored from the penalty spot as the Blue Samurai skipped away to a four-goal lead by the 17th minute before Leicester City striker Chris Wood pulled two back for the All Whites either side of half-time.
Shinji Okazaki opened the scoring for Japan, supplied by Kagawa, before the United man doubled the lead. Masato Morishige and Okazaki added more goals before the 20-minute mark but still Kagawa wasn't completely happy.
"We were able to get an early goal so that helped me get into the game. I'm happy I was able to earn the penalty and convert it," he told reporters.
"We all had tight schedules before the match but we'd talked about getting into the game and building our momentum quickly. [New Zealand] weren't able to get into the match quickly so it's great that we were able to score so early.
"We have to consider that we still gave up goals, and the goals we scored were thanks to our opponent's mistakes, so it's no question that our games at the World Cup will be much closer.
"In that sense we have a lot to improve on."
Japan next meet Cyprus in May before taking on Colombia, Ivory Coast and Greece in Group C at the 2014 World Cup in Brazil.
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