A Mario Mandzukic goal in the 109th minute sealed an extra time victory for Croatia after an early Kieran Trippier free kick was canceled out in the second half by Ivan Perisic during normal time.
"Until they scored [the equaliser] we were always under cosh", Kane said.
It was his first goal for his country, England's 12th of the tournament and ninth from a set-piece, but they should have added more from open play as they revelled in the space they were being given in the first half.
Gareth Southgate's men threw everything that they had at the Croatians, including - it turns out - an attempt to take advantage of a perceived rule loophole.
The midfielder fought through the illness to play the entire 120 minutes as his side came from behind to beat England 2-1 in extra time.
"Obviously it's been good to get to this stage and we've done everyone proud but we wanted to go on and win it. but we've fallen a bit short".
"We've left everything out there, you can look at the goals and the chances, but I don't think over the six or seven weeks we've been together we could've given any more".
England had been trying to reach its first World Cup final since winning its only title in 1966.
Thai boys lost weight but generally well after cave ordeal
Police officers took photographs of each other at the massive cave entrance, as pumps continued to suck out huge volumes of water. Narongsak Osatanakorn, the official overseeing the rescue operation , said the boys should not be blamed for their near tragedy.
It was a different game from then on, with Croatia suddenly playing with accuracy and goal as England lost their shape and composure, though Kane missed a chance to win it in stoppage time when he headed wide from a free kick.
For English soccer fans used to painful World Cup exits, the semifinal loss hurt.
Croatia's papers celebrated their history-makers - as well as suggesting, like Luka Modric did, that England had underestimated their talents.
Mario Mandzukic became Croatia's hero when he put Croatia in front with 11 minutes to go after a great through ball from Pivaric.
At the other end, Perisic had a shot on target blocked by Kyle Walker but it two minutes later there was no stopping Perisic as he volleyed volleyd home a cross from Vrsaljko on the right.
As a result, Croatia are through to their first ever World Cup final.
England captain Harry Kane says he believes his side "restored pride in the shirt" after the team's gallant march to the World Cup semi-final, which ended in a 2-1 defeat to Croatia at Luzhniki Stadium in Moscow.