Kane flicked in Djed Spence’s cross in first half stoppage time as England laboured to victory against a side 81 places below them in the world rankings.
“We’re here for preparation more than the result,” said Kane after his 67th goal of the season for club and country. “It’s the best shape I’ve ever been in my career.
“I’m excited, can’t wait for it to get started.”
New Zealand were thrashed 4-0 by Haiti on Wednesday but Thomas Tuchel’s men found clear-cut chances far harder to come by in their first experience of the brutal conditions in the United States.
Tuchel named two entirely different sides for each half, but there was little to read into his team selection ahead of England’s opening game of the World Cup against Croatia on June 17.
“The energy is right and the quality will come,” said Tuchel. “The better the opponent gets, the better we will get.”
Kane typically made the breakthrough from an in-swinging cross by Spence, who impressed at left-back.
Ivan Toney replaced Kane among the 11 changes made by Tuchel at the break.
Toney thought he had won a penalty only for New Zealand to be rescued by a marginal offside call against the Al Ahli striker.
England dominated possession but could not add to their lead with the Kiwis happy to keep the score down.
Despite not having won a major tournament for 60 years, England go into the World Cup among the favourites having reached at least the quarter-finals in their last four tournaments.
They also face Ghana and Panama in Group L.
New Zealand will face Iran, Belgium and Egypt in Group G.
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