Thomas Tuchel's charges twice took the lead in the first half only to be pegged back.
Kane opened the scoring in the 10th minute from the penalty spot. The Croatia goalkeeper Dominik Livakovic saved Kane's kick but was deemed to have strayed from his goalline before the ball was struck.
Kane buried his second chance into the bottom right hand corner of the net.
Martin Baturina levelled after 36 minutes with a sumptuous shot past the England goalkeeper Jordan Pickford into the top left hand corner of the goal.
As half time approached, Kane restored England's advantage after heading the ball into the net from a Declan Rice corner kick.
It was his 10th goal for England at his third World Cup to draw level with Gary Lineker as his country's most prolific striker at the tournament. But Kane's joy was cut short.
With virtually the last action of the half, Petar Musa drew Croatia level again.
High energy return from half time
England emerged from the pause and maintained the high energy tempo.
Within two minutes of the restart, Jude Bellingham scored England's third goal following a muscular run down the right wing and then sliding the ball past Livakovic.
Substitute Marcus Rashford added a fourth goal five minutes from time to secure the points.
“It was a game of two halves," Kane told British broadcaster ITV. "First half we were OK. It was really disappointing to concede in the way we did.
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"But we went full gas in the second half and they couldn't deal with it.
"That's the level we have to set for every game. Credit to everyone, the intensity we went at is our biggest strength and we have to use that more in this tournament.
"It's three points on the board and a great start."
England will play Ghana in Boston on 23 June while Croatia will attempt to relaunch their World Cup ambitions against Panama in Toronto on the same day.
All four teams conclude their Group L campaigns on 27 June.


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