Noni Madueke won an early penalty after a challenge from Luka Modric, and Kane converted at the second attempt after his initial effort was saved, with the kick retaken for encroachment.
Martin Baturina levelled for Croatia before Kane restored England's lead with a header from a Declan Rice corner, taking him level with Gary Lineker's England record of 10 World Cup goals.
Petar Musa equalised again for Croatia in first-half stoppage time, finishing after a headed pass from Ivan Perisic.
Jude Bellingham put England back in front just two minutes into the second half, driving into the box from the right before finishing low into the far corner.
Substitute Marcus Rashford added a fourth in the closing stages, combining with fellow replacement Bukayo Saka to seal the win.
England forward Anthony Gordon, making his World Cup debut, reflected on the occasion after the final whistle.
"It has been a crazy couple of weeks and that just topped it off. First World Cup game, something I have dreamed about as a kid."
Gordon was quick to turn the focus away from himself and onto the team.
"Special, but it is not about me. Self-centredness is a disease and I don't want to be a part of that. It is about the team. Rashy came on and made an impact, Bukayo and Morgan. It is a collective."
He also addressed the manner of the win, pointing to a much-improved second-half display.
"A difficult first half - their goal came from nowhere and stunned us a little bit. We came out really strong in the second half and got what we wanted. They were really good and that can't be underestimated when you look at the game."
Bellingham, whose goal proved decisive, spoke of his pride in helping the team after a difficult club season.
"For me personally it was nice to put some of the noise aside and show my country, show my teammates how committed I am to helping us win football matches."
He praised the team's overall performance, particularly after the break.
"I think it was a great team performance second half. First half we still got the intensity right but with the ball not quite there and second half I think we put it all together nicely."
Asked about his goal, Bellingham credited the build-up play.
"It was a good pass from Elliot and it put me in a good position and took it in the box. Pretty normal goal really apart from the pass was top level."
He also addressed criticism he has faced this season, and the backing of his head coach.
"It's been a tougher season for me. I feel fresh and sharp and strong and ready to give everything I have. It's nice to hear those kind of things."
Asked whether he had arrived at the tournament with extra motivation, Bellingham did not hold back.
"A little bit. I've got a little bit of a chip on my shoulder. I think that helps me a lot to find that focus early in the game, to find that intensity."
England head coach Thomas Tuchel admitted the nature of the win left him both pleased and relieved.
"Yes, happy and also relieved. Of course, because there is a lot of pressure on the first match and you want to have a good start. It is a big opponent and a top 10 team in the world."
Tuchel was critical of his side's decision-making during a difficult opening 45 minutes.
"When we could have played short we played long and when we could have played long we played short. We were not precise in our passing. We were slow in our decision-making sometimes."
He pointed to a lack of defensive cohesion as a factor in Croatia's two goals.
"Even off the ball, we were not connected as we should be. We didn't get the press 100% right. And also we were leading twice, and it felt like we now had something to protect, and we were always stepping back to protect something which is also not us."
Tuchel was encouraged, however, by his side's response after the interval.
"I am very happy about the result and also the second half, because we showed a good reaction and the players went for it and played with courage and played with intensity, and we played on the front foot. We created a lot of chances and scored some great goals."
England top Group L after the opening round of fixtures, and will face Ghana in their next match.