Defeat means England's long wait for a trophy since 1966 goes on and the Dutch thoroughly deserved their place in Sunday's final against Portugal.
Marcus Rashford's penalty gave England a first-half lead against the run of play, but Matthijs de Ligt made amends for his error in leading to the spot-kick to level 17 minutes from time.
England contributed to their own downfall in extra-time with a comedy of errors for both goals. John Stones and Ross Barkley were at fault as Kyle Walker's own goal and Quincy Promes's simple finish sent Holland into the final.
Southgate's men were attempting to build on the success of reaching a first World Cup semi-final for 28 years, but their display showed much work is still to be done if they are to end their trophy drought on next summer at Euro 2020 with the semi-finals and final to be played on home soil at Wembley.
"I think it's a really important game for us to reflect on," said Southgate. "The next few days will be painful having got to this stage, but we will be stronger for the experience."
Southgate defended his decision to leave out all seven of the Champions League finalists in his squad from Tottenham and Liverpool as captain Harry Kane and Jordan Henderson were left on the bench.
And the England boss also insisted his side must persist with passing the ball out from the back despite the criticism that is bound to come Stones and Barkley's way.
"If we didn't play that way we would never progress to be a top team," said Southgate.
"That's the way I believe they can play and should play.
"The errors were uncharacteristic, it was just poor execution and fatigue. I've got to not overreact to those things and support them in a difficult moment for everybody."
- De Jong shines -
The Netherlands failed to even qualify for the last two major tournaments. However, the young talent that has seen a dramatic turnaround in their fortunes was in evidence with Barcelona-bound Frenkie de Jong outstanding in midfield.
"I heard he was player of the match tonight, well deserved," said Netherlands boss Ronald Koeman.
"Most of the people are always looking what he is doing with the ball, how calm he is, but if you see defensive wise how many balls he is winning in midfield. It looks so easy, but it isn't."
De Ligt also looks set to leave Ajax with a host of top European clubs after his signature, but he had a moment to forget just after the half-hour mark.
The 19-year-old failed to control a simple pass and then chopped down Rashford, who picked himself up to slot home the resulting penalty.
However, De Ligt showed great character to make amends by powering home a header from a corner as the Netherlands' pressure finally told.
"We dropped too deep and it took them to equalise for us to step it up again," lamented Kane.
Just when there seemed only one winner, England produced their best move of the game as Stones, Ben Chilwell, Barkley and Raheem Sterling moved the ball from one end to the other before Jesse Lingard slotted home.
The celebrations were cut short, though, when the Manchester United midfielder was ruled offside by the finest of margins after a VAR review.
Memphis Depay and Donny van de Beek blazed over fine Dutch chances to win the game before extra-time, but the Netherlands finally took complete control in the extra 30 minutes.
England inflicted much of the damage themselves as Stones dallied on the ball and after Jordan Pickford made a great save to deny Depay, Promes's follow-up deflected in off Walker.
Barkley's underhit backpass then teed up a third for the Dutch as Depay squared for Promes to slot into an empty net.