Ollie Watkins struck a stoppage-time winner in Dortmund to send England into Sunday's final against Spain in Berlin.
For the third consecutive game England were forced to come from behind after Xavi Simons' early thunderbolt for the Dutch.
But Southgate's men got their rewards for a much-improved performance as Harry Kane levelled from the penalty spot before Aston Villa striker Watkins proved the unlikely hero after his limited role at the tournament so far.
"I was really pleased with the quality of our play. It will be, I would imagine, a long time, if ever, an English side had 60 percent of the ball against a side from the Netherlands. So it shows the more modern England way," said Southgate.
The England boss had been heavily criticised after his side needed late equalisers against Slovakia and Switzerland to reach the last four.
But in four major tournaments in charge, Southgate has now led his country to two finals, a semi-final and a quarter-final.
And he is keen to make amends from losing the Euro 2020 final on penalties to Italy at Wembley three years ago.
"The only reason I did the job when I took it on was to try and bring success to England as a nation and to try and improve English football," he added.
"To be able to take the team to a first final overseas, I'm immensely proud of that.
"But now of course we've come here to win. We play the team who've been the best team in the tournament and we have a day less to prepare, so it's a huge task. But we're still here and we're fighting."
Victory was especially sweet for Southgate after the backlash he received for England's shaky start to the tournament.
He had been targeted by thrown beer cups after a 0-0 draw against Slovenia in the group stage.
"We all want to be loved, right?" Soughtate said on his warmer reception from the travelling fans in Dortmund.
"When you're doing something for your country and you're a proud Englishman and when you don't feel that back and when all you read is criticism, it's hard.
"So, to be able to celebrate a second final is very, very special."
