Kieran Trippier gave England the lead after just five minutes of play but Ivan Perisic and Mario Mandzukic struck to put the Vatreni back in front and into the World Cup final.
Kane was left to reflect on the “small margins” that proved the difference between victory and defeat but was also confident about the progress made by the Three Lions since their struggles at Euro 2016 as well as Brazil 2014.
“It’s been an amazing run but it hurts to lose, it really hurts,” Kane told BBC Sport. “We wanted to go on and win it and we thought we had enough to go through but it wasn’t to be.
“In big games, small margins make the difference and they had a ball that fell their way at the end and he stuck it away. We played well at the start and in extra time but we missed a few chances to kill them off.
“There are a lot of positives to take from this tournament. It’s a big stepping stone from two years ago.”
England had previously reached the 1990 World Cup semi-finals and they will now take on Belgium in Saturday’s third-place match in Saint Petersburg.