Kane scored twice as England thrashed Ukraine 4-0 to qualify in Rome for the last four, where they will face Denmark at London’s Wembley Stadium.
England bowed out of the 2018 World Cup at the semi-final stage, along with Euro 96 on home turf when Southgate himself had his spotkick saved in a shoot-out loss to Germany.
“We want to go two steps further,” Southgate said. “I know what will be happening at home. It’s lovely to see everyone on a Saturday night, beer in hand.
“They should enjoy it. It’s been a long year for everyone. I’m chuffed the two performances have brought so much happiness to people.
“It’s fabulous for our country - a semi-final at Wembley. Everyone can really look forward to that - it’s brilliant.
“We’re not satisfied. Tonight is another very enjoyable night for everybody but even before the end of the game, I was thinking about the next challenge.
“We have never been to a European Championship final. It’s another opportunity to make history.”
Kane had started the Euros slowly, failing to score in the group stage, but he netted against Germany in the last 16 before his brace against Ukraine.
“A lot of us playing now won’t have the opportunity to have a big game like that at Wembley in a major tournament again,” Kane said.
“It’s there for us. It’s an opportunity for us and we have to grab it with both hands.
“We have to enjoy being in that environment. Coming off the back of the Germany game, the spirits were high and the stadium was rocking so hopefully we can use that energy again and really take the game to Denmark.”