The Americans trailed 2-1 heading into Oudin's match against Betina Jozami, but the American battled to a 2-6, 6-1, 6-2 win to tie the best-of-five competition.
Ditty and Huber then beat Jozami and Gisela Dulko 6-2, 6-3 to win it for the Americans.
Earlier in the day, Dulko topped American Jill Craybas to give the Argentines a one-point advantage heading into the two final matches.
The United States team will go against the Czech Republic in the semifinals in April. The Czechs eliminated Spain Sunday.