Having trailed 2-1 overnight, Federico Delbonis was the eventual hero as he cruised to victory over Ivo Karlovic in the decisive rubber on Sunday evening at the Arena Zagreb.
Earlier in the day, Juan Martin del Potro had come from two sets down to beat Marin Cilic to force the fifth and final rubber.
The 2009 US Open champion looked dead and buried after being outclassed in the second set but fought back heroically to win 6-7 (4-7) 2-6 7-5 6-4 6-3 after almost five hours of play.
That left Delbonis and Karlovic to battle it out for the crown and it was the South Americans who prevailed to claim their maiden title, after previously losing in four separate finals.
Breaks in each of the opening two sets gave the world number 41 the advantage and he eventually ran out the 6-3 6-4 6-2 winner to give his country a 3-2 victory following a dramatic weekend in the Croatian capital.