The two sides failed to threaten during the opening stages as Chile stifled the attacking prowess of the Dutch.
However, the South Americans almost took the lead with 20 minutes played as Felipe Gutierrez fired high over the crossbar from the edge of the box following a clever corner routine.
The Netherlands were up the other end moments later, but Georginio Wijnaldum screwed his shot wide when he had plenty of time and space to do better.
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The Netherlands continued to show more attacking intent, and they were inches away from taking the lead late on in the half as Arjen Robben shot narrowly wide following a trademark run through Chile's defence.
Chile created the first meaningful opportunity of the second period when Sanchez blasted the ball over from 25 yards.
However, both sides continued to struggle in front of goal during what remained a cagey encounter between two physical nations.
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Substitute Memphis Depay had a powerful drive tipped over with 25 minutes left before the Dutch finally opened the scoring when a corner was headed in by Leroy Fer, who had only been introduced off the bench moments earlier.
Chile pushed hard for an equaliser as Marcelo Diaz and Sanchez both threatened, but they were unable to breach the Netherlands' solid defence.
Instead, the 2010 finalists sealed the win in added time through Depay, who tapped in from close range following a cross from Robben.