The world champions took a two-goal lead just before the hour in the international friendly, but Kane scored England's first, before Jamie Vardy and Eric Dier completed the turnaround for Roy Hodgson's side.
Kane refused to hide his delight when questioned on the performance, but insisted that "everyone needs to stay calm" ahead of the European Championships this summer.
"We played brilliant today, had a lot of chances, we could have scored more. We never gave in. What a night," Kane told ITV Sport.
"Everyone needs to stay calm. It is a friendly, but it is a great night for England and shows what a team we are. We have to keep coming back, keep doing what we're doing, we have another important game on Tuesday against the Netherlands and go from there."
Kane has now scored four times in nine appearances for his country.