Croatia started the final day of the match here on Sunday with 2-1 lead.
But Argentina's Juan Martin del Potro beat Marin Cilic 6-7(4), 2-6, 7-5, 6-4, 6-3 in a match that lasted four hours and 53 minutes, reports Xinhua.
Federico Delbonis from the Argentine team then swept past Ivo Karlovic 6-3, 6-4, 6-2 in the decisive fifth singles match to turn Argentina's long-time dream into reality.
Argentina had lost in the final of the tournament four times earlier while Croatia won the title in 2005.