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.
Diaz started for Colombia in Arizona as they impressed with a 5-0 defeat of Panama and the Liverpool forward delivered his country’s third strike of the night shortly before half-time.
Jhon Cordoba, James Rodriguez, Richard Rios and Miguel Borja were also on the scoresheet for Los Cafeteros, with Diaz withdrawn from proceedings in minute 65.
Luis Diaz had an excellent match when he continuously stirred up the opponent’s defense with 3 Dribbles and 11 Duels won.
Elsewhere in the quarter-finals, Nunez’s Uruguay met Alisson Becker’s Brazil in Las Vegas and both Reds started the contest, which ended 0-0.
Darwin Nunez had a difficult day competing as he constantly had to move to support his teammates and only had 4/11 duels.
Alisson made one excellent save during the subsequent penalty shootout but it was Uruguay who won out, 4-2, to advance.
Uruguay and Colombia will now face each other in the last four, with their meeting set for the early hours of Thursday morning.