The North Africans held their nerve during the penalty shootout, with Ismael Saibari converting the decisive spot-kick after goalkeeper Yassine Bounou made a crucial save to deny Crysencio Summerville.
Saibari calmly sent goalkeeper Bart Verbruggen the wrong way to seal Morocco's place in the knockout stage.
The result also marked the earliest FIFA World Cup exit in the Netherlands' history, while Morocco, who reached the semi-finals in the previous edition, continued their impressive run in the competition.
The Netherlands appeared to be heading for victory after Cody Gakpo opened the scoring in the 72nd minute.
The Liverpool forward finished clinically from a pass by Crysencio Summerville to put the Dutch ahead. Gakpo was visibly emotional after finding the net, with teammates and members of the coaching staff embracing him.
The forward recently revealed that he and his partner, Noa van der Bij, had suffered the loss of their unborn child. However, Morocco beat Netherlands after staging a dramatic late comeback.
In the first minute of stoppage time, Issa Diop headed home Chemsdine Talbi's cross to level the score at 1-1 and force extra time. Morocco nearly completed the turnaround in the 97th minute when Soufiane Rahimi raced through on goal, but Bart Verbruggen produced an outstanding save to keep the Netherlands in the contest.
Neither team managed to find a winning goal during the remainder of extra time, sending the match to a penalty shootout.
Morocco converted four of their penalties, while Bounou's decisive save proved to be the turning point as the Atlas Lions secured a deserved victory.
The North African side will now face Canada in the FIFA World Cup round of 16 on Saturday in Houston as they continue their quest for World Cup glory.
The encounter was the second match at the tournament to be decided by penalties after Paraguay eliminated Germany earlier on Monday.
Morocco and the Netherlands entered the fixture ranked sixth and seventh respectively in the latest FIFA World Rankings.
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