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.
Delbonis dropped to the ground after his 6-3, 6-4, 6-2 victory.
The fate of the squad did not look good going into the second day, when Croatia led 2-1 after Friday and Saturday.
Delbonis, who won the final match, said that "I still can't believe it". "Everything could have been worse; let's not forget we will be alive on a Sunday of a Davis Cup final".
Diego Maradona led thousands of visiting fans with chants and frantically waved a scarf with the Argentine flag from his executive booth at Arena Zagreb.
The Argentine saved three break points in the seventh game of the third set and claimed it with a searing forehand victor.
Argentina captain Daniel Orsanic added: "His comeback is the comeback of history, his comeback today was the comeback of the Davis Cup. Davis Cup series are always physically challenging, with long matches in which the mental part plays a key role, and I am ready to face the challenge".
He said the injury happened in the fifth set while he was trying to catch a ball.
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.
Having clinched the opening set tiebreak, Cilic raced through the second set before Del Potro, who improved his record against the Croatian to 9-2 in all competitions, started his comeback with a passing shot between his legs. "They deserve to win the tournament after eliminating the U.S. away and defeating France, but we made it really far after playing the whole year away from home".
"It is very hard to lose such an important match", Cilic said.
Trailing 15-40 in the eighth game of the final set, Cilic sent a forehand beyond the baseline to give Del Potro the decisive break. But one match after the other, Argentina had the team spirit it lacked on other occasions (exactly four) to conquer the most desired trophy in Argentine sports.