Kane scored an 85th minute winner after substitute Jesse Lingard had cancelled out Andrej Kramaric's opener, as England came from behind to win 2-1 and book their place in next summer's inaugural Nations League Finals.
The result also condemned Croatia to relegation, as England gained a measure of revenge for the World Cup semi-final defeat by the same opponents in Moscow this summer.
Wembley was in full voice behind Gareth Southgate's side's late recovery, and Kane was full of praise for the home support.
"It's incredible, I've never seen Wembley like this in an England shirt," he told Sky Sports. "We want to make everyone proud. I know they'll enjoy this one.
"We felt the crowd. We felt their presence. That's what we need everywhere we go. It felt good. The lads stuck at it, we showed our character, stayed calm and deserved to win."
The England captain admitted that his teammates are not yet sure just how big a triumph a Nations League victory would represent, but says the Three Lions will do everything to become the first team to lift the trophy in Portugal next summer.
"I said the other day, nobody really knows how big this competition is going to become," he said. "You don't really get many opportunities to win a tournament in an England shirt, so we have got to try and take full advantage of that. And it's another semi-final for the fans to look forward to as well.
"We are professionals, we want to win everything we do, whether it is a new tournament or an old tournament we want to win. It's a chance to win a trophy in an England shirt so we are going to try and do that."