After 120 goalless minutes, the Albiceleste won in a shootout to book their place in the final against Germany
Argentina reached the World Cup final as they beat Netherlands in a penalty shootout on Wednesday.
Having introduced goalkeeper Tim Krul off the bench to replace number one Jasper Cillessen in the quarter-final shoot-out win over Costa Rica, Louis van Gaal had no such options this time around as Wednesday's semi-final finished 0-0 after extra time.
The Dutch had used all three substitutes by that point, leaving Cillessen to face Argentina's efforts when it mattered most.
And the youngster, who was playing in the second tier of Dutch football less than a year ago, failed to reproduce his team-mate's heroics as opposite number Sergio Romero saved from Ron Vlaar and Wesley Sneijder to book Argentina's place in the showpiece against Germany on Sunday.
Ultimately, Maxi Rodriguez sealed a 4-2 shoot-out success, crashing his penalty in off the crossbar via the palms of Cillessen to spark jubilant scenes among the South Americans in the stands in Sao Paulo.
In a contest that was a far cry from the stunning drama of Germany's humiliation of Brazil 24 hours earlier, penalties always seemed the most likely outcome and so it proved.
The last time these two teams met in the knockout stages of a finals - in 1998 - the game was lit up by the sublime quality of Dennis Bergkamp's winning goal, but such moments of outstanding individual brilliance were almost non-existent here.
Much of the build-up was dominated by talk of a battle for supremacy between the two sides' respective talismen, but both Lionel Messi and Arjen Robben were well shackled throughout.
With many neutrals having hoped for a Brazil-Argentina final, the capitulation of the hosts on Tuesday meant that was a distant dream, but the success of Alejandro Sabella's men at least ensures that there will be South American representation in the showpiece.
The Dutch, runners-up four years ago, will now face a wounded Brazil in the third-place play-off in Brasilia on Saturday.
Netherlands made one change from their quarter-final win over Costa Rica as Nigel de Jong - who it was thought had been ruled out of the tournament with a groin strain - made an unexpected recovery to return in place of Memphis Depay.
For Argentina, Marcos Rojo came back from a one-match suspension, and Enzo Perez earned a spot in the absence of the injured Angel di Maria.
Neither side carved out a clear goalscoring opportunity in a turgid first period, Sneijder firing an effort on goal in the 13th minute, but his volley from outside the box was comfortably wide of the left-hand post.
Messi threatened for the first time two minutes later, seeing his free-kick well held by Cillessen diving low to his right.
Argentina looked marginally the more dangerous of the two sides and midway through the first half Ezequiel Garay headed Ezequiel Lavezzi's left-wing corner over the crossbar under pressure from Vlaar.
There was a moment of concern for Argentina shortly afterwards, when a clearly dazed Javier Mascherano fell to the floor following a clash of heads with Georginio Wijnaldum.
After receiving treatment, Mascherano was able to walk off the field and soon returned to action.
The second period began in much the same fashion as the first, with neither side able to make any inroads in the final third of the field.
Gonzalo Higuain did have half an opening in the 58th minute, but substitute Daryl Janmaat's covering header denied the Napoli man the chance to get on the end of Lavezzi's delivery from the right.
Once again, Higuain was unable to capitalise on a rare sight of goal 15 minutes from time, stretching to divert Perez's clever low ball into the side netting from close range.
Rojo tried his luck from distance nine minutes later, but his well-hit drive was straight at an untroubled Cillessen.
Robben's big moment arrived in the first minute of stoppage time as he bore down on goal following a fortuitous ricochet on the edge of the area, but Mascherano recovered brilliantly to deny the Bayern Munich man with a last-ditch tackle.
The winger had the first effort of note in extra-time, bringing a routine save out of Romero with a dipping effort from 25 yards after a trademark darting run off the right flank.
There were two late chances for Sabella's men in the final five minutes of the extra period, but both Rodrigo Palacio and Rodriguez failed to gain sufficient power on their efforts to really test Cillessen.
But, with Krul watching on from the sidelines this time, there was little that the young Dutch goalkeeper could do to prevent a clinical Argentina booking their place in the final when the spot-kicks arrived, as first Vlaar and then Sneijder proved wayward before Rodriguez struck the decisive blow.