Netherlands had to dig deep to send the game to extra time after falling two goals behind at the Lusail Iconic Stadium.
Lionel Messi was once again at the heart of everything good Argentina did; first setting up Nahuel Molina's opener in the 35th minute before dispatching a penalty in the 73rd minute after Manuel Acuna had been fouled by Denzel Dumfries in the box.
But Netherlands refused to let their heads drop as substitute Wout Weghorst glanced home a cross with seven minutes of normal time remaining.
The Besiktas striker was again on hand to send Dutch fans into raptures as he fired home a neatly worked free kick in the 11th minute of stoppage time.
Another 30 minutes of enthralling action could not produce any more goals and penalties were needed to separate the two sides.
Aston Villa goalkeeper Emiliano Martinez proved to be the hero for Argentina as he saved Netherlands' first two spot kicks from Virgil Van Dijk and Steven Berghuis, leaving Lautaro Martinez to fire home the decisive spot kick to take the South Americans through.
Argentina also knocked the Dutch out in the semi-finals stage on penalties at the 2014 FIFA World Cup in Brazil.
Lionel Scaloni's side will next meet Croatia for a place in this year's final.
