Argentina had one foot in the semi-finals courtesy of strikes from Nahuel Molina and Lionel Messi, but Wout Weghorst came up with two remarkable goals off the bench - including one in the 10th minute of second-half injury time - to keep the contest alive in the most dramatic of circumstances.
No more goals were to be had in the additional 30 minutes, but the Copa America winners held their nerve from the spot to win 4-3 in the shootout and set up a semi-final with Croatia.
To say that the opening exchanges were cagey would be an understatement, as the Netherlands enjoyed plentiful possession but failed to turn it into anything meaningful.
Seven-time Ballon d'Or winner Messi was unsurprisingly central to Argentina's forays forward, and his everlasting excellence helped La Albiceleste take the lead with 35 minutes gone.
Messi's twinkle toes lit up the Lusail Iconic Stadium before the Paris Saint-Germain attacker threaded a ball through to right-back Nahuel Molina, who fended off the attention of Virgil van Dijk to poke home into the far corner.
Argentina suddenly had their tails up, and the Netherlands could not answer the questions posed to them by a new-look five-man Argentina backline, leading Louis Van Gaal to bring on Teun Koopmeiners and Steven Berghuis for Steven Bergwijn and Marten de Roon at half time.
Oranje remained toothless in the final third, though, and Argentina were more than happy to soak up the pressure before Messi won a free kick in a dangerous area in the 63rd minute, but the 35-year-old sent his strike a whisker over the bar.
Van Gaal's side would be made to pay for their passivity in the 73rd minute, as Denzel Dumfries tripped Marcos Acuna just inside the area to hand Messi the chance to double his tally from the spot, and the veteran left Noppert rooted to the spot with a perfect penalty into the side of the net.
Messi's 11th World Cup goal saw him surpass Gabriel Batistuta as Argentina's leading goalscorer of all time in the tournament, but the Netherlands would give themselves a glimmer of hope out of nowhere in the 83rd minute, as substitute Weghorst met Berghuis's delivery from the right to guide a header past Emiliano Martinez.
Van Gaal's side finally demonstrated some attacking intent that was absent during the previous 80 minutes, as Berghuis smashed an effort into the side netting in the 85th minute during his side's desperate search for a leveller.
Tensions soon boiled over in the 89th minute when Leandro Paredes booted a ball into the Dutch dugout after fouling Nathan Ake, enraging the Netherlands contingent and leading to a brief melee, but card-happy referee Antonio Mateu did not banish anyone from the field.
The Netherlands were handed a 10-minute period of added time to try to restore parity, the final one of which saw Van Gaal's side handed a last-gasp free kick in a promising position, and Weghorst made the seemingly impossible possible.
Rather than go for goal, Koopmeiners played a delicate pass into the box for Weghorst, who took a touch and managed to prod home to miraculously force another 30 minutes, prior to which the two sets of hot heads immediately clashed once again.
For all the fiery encounters off the ball, neither nation provided a spark in the opening period of extra time, but in the 114th minute, Van Dijk had to stand firm to keep out Lautaro Martinez's powerful drive after some strong play on the right from Angel Di Maria.
Only one minute later, a long-range strike from Enzo Fernandez landed on the top of the net after a significant deflection, and the Benfica youngster's fizzed effort then struck the post as injury time approached.
Penalties would be needed to decide a marvellous quarter-final, and it was advantage Argentina after the first spot kick, as Emiliano Martinez sprung to his right to keep out Virgil van Dijk's effort.
The perennially calm Messi converted his penalty to put Argentina 1-0 up, and Martinez then dived to his left to keep out Berghuis before breaking out into a celebratory dance.
Fernandez had the chance to seal Argentina's place in the final four, but he dragged his effort wide, and Luuk de Jong sent Martinez the wrong way to keep the Dutch dream alive.
Lautaro Martinez had to score La Albiceleste's fifth and final penalty to end the contest once and for all, and the Inter Milan striker kept his composure to send his nation into the semi-finals.