Kane was left to reflect on the "small margins" that proved the difference between victory and defeat, but was also optimistic about the progress made by England since their struggles at Euro 2016 and 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, who previously reached the semi-finals of the World Cup in 1990, will now play Belgium in Saturday's third-place playoff in St Petersburg.