Morgan Rogers gave Thomas Tuchel's side the lead before Petar Suฤiฤ took advantage of a defensive mistake to bring Croatia level. Kane then converted from the spot to settle a demanding contest and give England an important result ahead of Sunday's meeting with Spain.
Tuchel was pleased with the response from his players after a difficult opening spell.
"Good win. We wanted this win desperately and I think we deserved it. I think it was hard work against a good opponent. It took us a while to grow into the game but after 15 minutes I think we did OK, got the goal and got better and more confident and had some good actions and good chances, good half-chances.
"Got a little bit sloppy at the very end of the first half, started the second half very good, didn't score the second goal, conceded from a set piece, and then showed a good reaction.
"We got the job done and it was important because it was a special occasion for Harry [Kane] and he scored the winner, so perfect evening."
The England manager felt his side had been close to extending their advantage before allowing Croatia back into the match.
"It was maybe only three or four minutes before we gave passes away and especially in a moment when we were clearly the better team, we came with a very strong reaction after the half-time, had a beautiful chance created from Declan Rice and they blocked it on the line.
"It was like our third goal was coming and we had some easy ball losses in our own half that killed our momentum. An awkward moment in our box can always happen. It's important to react and we did it."
Tuchel was also satisfied with the work of his central defenders, Dan Burn and Jarell Quansah, against Croatia's attacking shape.
"Good. It was demanding for Dan Burn and also for Jarell [Quansah] because they [Croatia] play with a back five and have two number 10s in between the lines.
"It was a lot of work to do, a lot of communication but they did very well."
England now turn their attention to Spain, with Tuchel expecting a different tactical challenge.
"I don't know. Everything will be different. Completely different scenario, completely different setup, but I'm very positive that we will get what we want."
The victory keeps England's Nations League campaign on track and gives Kane a fitting conclusion to a landmark night at Wembley.