Substitute Ollie Watkins struck in the 91st minute to send England through to their second successive final.
Southgate said after the 2-1 triumph: "We all want to be loved, right?
"When you are doing something for your country and you are a proud Englishman, when all you read is criticism it is hard. To be able to celebrate a second final is very, very special.
"We have given people some amazing nights, some of the best nights in 50 years
"It shows the more modern England way but also the resilience and character of the group. Ollie Watkins has trained like that every day. He has trained for his moment no matter how frustrated.
"They had each others' back, they bonded so well and tonight it was an example of that."
On facing Spain in the final, Southgate continued: "We will have to get the ball off them first.
"It is not as simple as us having the ball and making them run. We have to be exceptional with the ball and without it. They have been the best team.
"We are starting to show a better version of ourselves.
"But we are in there with what we have shown to this point. We have as good a chance as they do. We have come here to win it. It is a huge task but we are still here and fighting."