With the shootout tied at 2-2 after four rounds, Morocco goalkeeper Yassine Bounou made a strong save of Crysencio Summerville's attempt, batting it away with his left hand.
Saibari then sent the winner into the low left corner as goalkeeper Bart Verbruggen went the other direction.
The Netherlands had reached at least the round of 16 in eight previous World Cup appearances, including a quarter-final appearance four years ago, when Morocco reached the semi-finals.
Cody Gakpo scored in the 72nd minute for the Dutch. After the goal, which was assisted by Summerville, the Dutch bench ran onto the field to embrace the 27-year-old Gakpo, who broke down in tears.
Gakpo and his partner, Noa van der Bij, recently announced they had lost their unborn child and he had decided to stay at the tournament.
Morocco's Issa Diop tied it in the 91st minute, heading in a long cross from Chemsdine Talbi after Dutch captain Virgil van Dijk watched the ball sail over his head.
The African nation had the best of the chances in 30 minutes of extra time but had goalkeeper Berbruggen to thank at Estadio BBVA.
It was the second game of the tournament to conclude with a shootout. Paraguay beat Germany earlier on Monday.
Moroccan striker Saibari has been prolific this tournament, also scoring a goal each against Brazil, Scotland and Haiti in the group stage.
The teams entered with the highest combined ranking of any round-of-32 match. Morocco were sixth in the world and the Netherlands seventh.
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