Del Potro, torn between what is best for his career and his desire to play for his country, had considered pulling out of the tie at Parque Roca because of pains in his left wrist but he managed to hold out for the win.
"How I suffer playing, just for you, a lot for you, I played to thank the fans for their affection, the support they give me," the Olympic bronze medallist, choking back the tears, told the crowd.
"The important thing is for me to be here... We all have pains, more or less bad... I know the one up there is looking after me," added the world number eight after the match played to a backdrop of raucous chanting by the home crowd.
Stepanek, who won in three of their four previous meetings, said he did not see his opponent struggling at any point in the match and suggested the issue of Del Potro's wrist might have been a bit of pre-match gamesmanship.
Del Potro had to save four break points in the eighth game before he broke the Czech. He gave away one break point in the second set but broke back in the next game and again in the seventh.
In the decider, Del Potro broke in the first and fifth games and settled the rubber on his serve with his first match point when Stepanek went wide with a return.
Czech world number six Tomas Berdych faces Juan Monaco in the second singles on the clay court at the Mary Teran de Weiss stadium.
The winning team will meet holders Spain or the United States in the final. Argentina will be looking to reach their second consecutive final, fourth since 2006 and fifth overall.
The Czechs, finalists in 2009, have won the trophy once as Czechoslovakia in 1980.