Argentina goalkeeper, Emiliano Martinez was the hero as he stopped two penalties to take his side to the semi-finals, where they will face Croatia.
Argentina were two goals up with quarter of an hour to go through goals from Molina and Messi.
But, the Orange Gang fought back to equalise and took the game to extra time.
Virgil van Dijk and Steven Berghuis missed their spot kicks for Netherlands.
Argentina will take on Croatia, who also eliminated Brazil on penalties in the semi-final on Tuesday.
