Goals from Nahuel Molina and Lionel Messi had fired La Albiceleste into a 2-0 lead, before Netherlands substitute Wout Weghorst netted twice in the final ten minutes, ensuring extra-time would be played.
The tie was eventually resolved on spot-kicks, with Virgil van Dijk and Steven Berghuis having their shots saved by Emilio Martinez, before Inter Milan forward Lautaro Martinez smashed home the decisive penalty to send Argentina through to the semi-finals.
Having set up a mouth-watering clash with Croatia on Tuesday evening (19:00 GMT), Martinez's nation will be bidding to reach a first World Cup final since 1990.