The Three Lions reached the final four of the competition for the first time since 1990 but, despite taking the lead against the Vatreni, were pegged back for a 2-1 defeat after extra time.
Kane, currently the leading goalscorer at Russia 2018, conceded that Croatia "played better" on the night but is excited about the foundations built within the squad with manager Gareth Southgate.
"It is tough, we are gutted, we worked so hard and the fans were amazing," the England captain told ITV Sport.
"It was a tough game, a 50-50 game, I'm sure we will look back and there is stuff we could have done better. It hurts, it hurts a lot, but we can hold our heads up high - we have gone further than a lot of people thought we would do.
"We created some good chances being 1-0 up, there are lots of ifs and buts and it is hard in these games, it is small margins and it went their way today. Until they scored we were always under the cosh. There is a lot we could have done better but they played better.
"Obviously it has been great to get to this stage but we wanted to win. It shows we can be up here, that we can win knockout games, get to semi-finals and the next step is to go one step further. This is a great foundation we have built with the gaffer, we are proud of what we have achieved but we wanted more and are sad we couldn't give our fans a final."
England will now play Belgium in Saturday's third-place playoff in St Petersburg.