Del Potro had defeated Marin Cilic in 8 of 10 previous meetings.
Cilic won Friday's opening singles against Federico Delbonis and paired up with Ivan Dodig to give Croatia a 2-1 lead in Saturday's doubles, but Del Potro's stunning recovery kept alive Argentine dreams of a first title.
Del Potro had defeated Cilic in 8 of 10 previous meetings, but Sunday's clash was the first since 2013 and the Croat was invigorated by an animated Zagreb Arena crowd including Argentine football legend Diego Maradona.
Sixth-ranked Cilic dominated a first set tie-break, winning the first five points to seize control, and then broke Del Potro's serve twice in succession in the second set to move Croatia within sight of a second title.
But Del Potro, who rallied from behind to overcome Andy Murray in a five-set epic in the semi-final, displayed more remarkable resilience and produced an outrageous 'tweener' to begin the third set.
Cilic fought off two early break points but then succumbed to nerves as the finish line approached, falling 0-40 behind on serve at 5-6, and Del Potro pounced at the third opportunity to reignite his country's hopes.
Del Potro's blistering forehand piled the pressure on Cilic, and the Argentine sent the match to a fifth set by converting his third set point after his opponent again faltered on serve at 5-4.
Del Potro committed a costly double fault to gift Cilic the advantage at the start of the deciding set, but the world number 38 hit straight back in the following game to level.
The Argentine then conjured up a pair of break points at 4-3 with a miscued forehand from Cilic paving the way for Del Potro to complete a stunning fightback -- his first ever from two sets down -- in four hours and 53 minutes.
Ivo Karlovic, 37, is scheduled to play 41st-ranked Delbonis with the title on the line as Croatia look to emulate their 2005 triumph while Argentina bid for a maiden crown at the fifth attempt.