Van Dijk led out his country on Saturday evening and completed the full contest alongside his Liverpool teammate, with Ryan Gravenberch an unused substitute in Berlin.
Gakpo played a decisive role in his nation’s victory, having initially been credited with the winner, before it was later altered to a Mert Muldur own goal.
Liverpool’s No.18 darted into the box to meet a low ball at the back post, which led defender Muldur to turn the ball into his own net with a sliding challenge.
Turkiye had earlier taken the lead in the opening half via Samet Akaydin, though saw their advantage disappear in six second half minutes after Stefan de Vrij’s header initially levelled the game in the 70th minute.
After securing victory, the Netherlands will now face England in the last four of the competition on Wednesday evening (kick-off 8pm BST).
Meanwhile, Trent Alexander-Arnold scored the winning spot-kick as England beat Switzerland on penalties to reach the semi-finals of Euro 2024.
Liverpool’s No.66 stepped off the bench in extra-time for his nation as the Three Lions progressed into the final four of the competition, with Reds teammate Joe Gomez an unused substitute in the tie.
In an even affair in Dusseldorf on Saturday, it was Switzerland who took the lead in the 75th minute via Breel Embolo’s finish inside the six-yard box.
England, however, levelled just five minutes later thanks to Bukayo Saka’s well-placed low strike from outside the area to send the game into extra-time.
With the two teams unable to be separated following the added half an hour, penalties were required, and it was Alexander-Arnold who converted the decisive effort from 12 yards to help his country to a 5-3 shootout victory.
Alexander-Arnold told BBC Sport post-match: “Incredible. They’re the goals and the aims that we set for ourselves. [They’re] a difficult opponent. Again, going behind, [but] the team has got a lot of character and a lot of belief and a lot of heart and spirit.
“We dug deep when it mattered most and [we] obviously get the win in the penalty shootout. We knew it was going to be tight. We knew we were never going to run away with the game, although that would’ve been nice. Whatever it takes, no matter what, we win and that’s all that matters to us.”
On his winning penalty, he added: “It’s what we practise. A lot of graft goes into that moment. They’re moments that I enjoy. When the gaffer says I’m taking one, the belly doesn’t drop.
“I enjoy it, I practise it. I knew what spot and I knew I just needed to execute it and I went and did that. All five penalties from us were great, Pickers [Jordan Pickford] with an amazing save to start the shootout gives you that confidence.
“We practise for these. You see every game so far has gone to extra-time in the quarter-finals. They’re tight margins, it was the finest of margins today, but we’ve been able to win.”