The 2-1 result sees England reach the Nations League semifinals.
Southgate said of Tottenham striker Kane: “Harry's the best goalscorer in the world, frankly. We've huge belief in him. He's so hungry to lead the team on and further.
“The biggest thing is the connection with fans – I've not seen the new Wembley like that. The way it feels with the fans is special. It's giving us the energy to try to go on and break new barriers all the time."
Southgate, whose decision to stick by the Spurs striker has been queried at times, added: “You're always loathe to take a player of Harry's ability off. He played a fantastic pass for Raheem Sterling after dropping off deep and turning in the first half.
“I know people have questioned some of my decisions, but Harry's our main penalty taker, our leader. In 98 per cent of games, his hold-up play and goalscoring is critical."