The second semi-final was a tamer affair than Germany's 7-1 annihilation of hosts Brazil, with star players Lionel Messi and Arjen Robben failing to light up the game.
All the drama came in the shoot-out, when Argentina goalkeeper Sergio Romero saved two Dutch penalties before Maxi Rodriguez converted the winning kick.
Sunday's encounter will be a repeat of the 1986 and 1990 finals.
If Germany win they will be the first European side to triumph at a World Cup staged in the Americas.
Meanwhile, Brazil will face Netherlands in Saturday's third-place play-off.
Away from the pitch, there had been fears of more protests over the cost of the competition following Brazil's humiliating exit on Tuesday - but so far the streets have remained peaceful.
Some ire has been directed at President Dilma Rousseff, who is seeking re-election in October.
Ms Rousseff, who had enjoyed something of a World Cup bounce as the team advanced during the competition, has said she is "very sad" about the result.
Ill feeling will not be helped by Argentina beating the Dutch to reach Sunday's final at Brazil's iconic Maracana.
The build-up to the competition was marred by rioting by those who thought the estimated $11.3bn (£6.6bn) should have been spent on public services and infrastructure.
Several buses were set on fire and looters ransacked an electronics store in Sao Paulo following Brazil's crushing defeat, although police did not confirm whether the trouble was linked to the match.
"It's a disaster. It will be chaos. People will break everything. They're going to be furious," said Karina Marques, a 17-year-old footballer who watched the game at a street screening in Rio attended by 30,000 people.
In tears, Maria Jose Costa Almeida, 35, asked: "Why spend so much on stadiums, bring the cup to Brazil, to win nothing?"
Antonio Hipolito, who lives in Rio, said: "I hope this can make people wake up and start thinking with their heads and not their emotions and that people translate the anger they are feeling at the ballot boxes.
"Soccer is just an illusion and we need to wake up to reality."
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