Victory to world number four David Nalbandian and Agustin Calleri over Paul Hanley and Wayne Arthurs 6-4 6-4 7-5 put the Argentines, losing finalists in 1981, into December's final.
Russia are currently 2-1 up in the other semi-final in Moscow after twins Bob and Mike Bryan beat Mikhail Youzhny and Dmitry Tursunov 6-3 6-4 6-2.
Nalbandian landed the first blow for Argentina when he defeated Mark Philippoussis in the opening singles on Friday and Jose Acasuso finished off a see-saw five-set win over Lleyton Hewitt earlier on Saturday.
The second singles rubber needed to be finished over two days after it had been suspended due to rain and fading light on Friday evening with Acasuso 4-0 up in the fifth set.
"I've beaten higher-ranked players in the past but to win at home in a Davis Cup semi-final with 14,000 people watching me here...makes it one of the most important wins of my career," Acasuso said after completing the 1-6 6-4 4-6 6-2 6-1 marathon victory.
Former world number one Hewitt believed the turning point came early in the fourth set when he squandered three break points and had his serve broken in the next game, Acasuso followed that up with eight games in a row.
"It was not taking those half chances which made the difference," he said.
In Moscow, the Bryan brothers kept the U.S. in the tie after Andy Roddick had lost to Marat Safin and James Blake had been beaten by Youzhny in Friday's singles.
"We had Andy and James coming up to us, saying 'Get us a chance, just give us another shot'," Mike Bryan said.
In Sunday's reverse singles, Roddick faces Youzhny before Blake takes on Safin.