Oranje were denied a repeat of the 2010 World Cup final against Spain courtesy of a gut-wrenching 2-1 defeat to England at the Signal Iduna Park, where the Dutch went ahead through a brilliant Xavi Simons strike.
However, England reacted well to going a goal down and were level just 11 minutes after Simons's thunderbolt, as Kane's perfect penalty found the bottom corner despite the best efforts of Bart Verbruggen.
Debate raged over whether Gareth Southgate's men should have been awarded the spot kick, though, as referee Felix Zwayer initially waved away the appeals after Denzel Dumfries caught Kane's boot as the Bayern Munich man volleyed over the bar.
Zwayer was sent to the pitchside monitor to review the decision, though, and the on-field referee ultimately pointed to the spot, much to the bewilderment of the Netherlands as Dumfries picked up a yellow card.
Virgil van Dijk echoes Koeman penalty sentiment
Kane's spot kick preceded a breathtaking 90th-minute winner from Ollie Watkins after a second half where the Netherlands were arguably the better team, and Koeman unsurprisingly raged at Zwayer's decision to point to the penalty spot in the first half.
"What do you do as a defender? This is not a penalty. Dumfries wanted to block the ball. Then the shoes collide. Such VAR decisions destroy football," the Dutchman said in his post-game press conference.
Koeman's words were echoed by captain Virgil van Dijk, who revealed that Zwayer made a dart for the tunnel rather than sticking around to shake his hand, which the Liverpool man thought was rather telling.
"The referee ran back inside straight after the full-time whistle, that says a lot. I had no time to shake his hand. But it is what it is, the game is done, we lost, certain moments were obvious that they should have gone our way, but they didn't, whatever the outcome," Van Dijk added.
Ex-England and Manchester United right-back Gary Neville sympathised with Oranje's frustrations, claiming that the award of the spot kick was a "disgrace" while doing punditry duties for ITV Sport, saying: "At any time, in a game of such importance, I was fuming by the Denmark one [a handball against Germany], I was offended.
Kane jokes 'my boot was hanging off' after Dumfries foul
"It's natural to block the shot, it's not a penalty, nowhere near a penalty. We're all going mad, we don't know what a penalty is. We've conditioned ourselves in the game to think that's a penalty.
"Honestly, that is not a penalty. I think in the box, there's a bit of licence to go in and block the shot. As a defender I think it's a disgrace."
However, England skipper Kane - who was rolling about in agony after Dumfries's challenge - joked that the Inter Milan right-back had taken his foot off and happily accepted the rub of the green.
"My foot's hanging off so he definitely caught me. Sometimes you get them sometimes you don't. I was happy to step up and see that go in the back of he net. It was a nice feeling for sure," Kane told the media.
England now have three full days to recover before heading to the Olympiastadion in Berlin to meet Spain in Sunday's Euro 2024 final.