The 19-year-old del Potro gave Argentina a 2-0 lead after seventh-ranked David Nalbandian beat Russian Igor Andreev 7-6 (5), 6-2, 6-4 in the first singles match.
Del Potro - who rose to 13th in the rankings from 81st since April after a 23-match winning streak - was playing his first home Davis Cup match. He showed no signs of nerves in front of the raucous sellout crowd of 14,000, making Davydenko hustle for nearly every point.
Davydenko made errors at the end of several long rallies, and Del Potro broke his serve in the fourth and sixth game to secure the first set. The Argentine then took a 4-1 lead in the second and was able to hold on after Davydenko managed a break of his own.
The Russian was broken again in the first and fifth games of the third set.
"I had a great debut and I hope we win one more point and get to the finals,'' del Potro said. "The support of the crowd made some really nerve-racking moments easier.''
Nalbandian won his first set after rallying from 4-1 down in the tiebreaker, and then got two breaks in the second. The third set went with service up to 5-4, before Nalbandian broke to clinch the match.
Nalbandian has a 26-6 Davis Cup record, and has never lost a home match since his debut in 2002.
In doubles Saturday, Nalbandian and Guillermo Canas are set to face Russia's Igor Kunitsyn and Dimitry Tursunov.
Agustin Calleri had been scheduled to play doubles with Canas, but Argentine captain Alberto Mancini changed the pairing after Nalbandian's singles match wasn't as physically demanding as expected and Mancini was more comfortable with the doubles pairing he'd used in recent Davis Cup matches.
Argentina has not lost at home in 10 years, winning 12 straight ties, but has never won the Davis Cup.
Spain is leading defending champion United States 2-0 in the other semifinal.
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Associated Press writer Eric Weil contributed to this report.
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