Delbonis dropped to the ground after his 6-3, 6-4, 6-2 victory. His teammates fell into each other's arms and celebrated in front of thousands of Argentine fans in the packed Arena Zagreb.
Players danced around the huge Davis Cup trophy and fans, including soccer great Diego Maradona, chanted and waved Argentinian flags.
This was the first Davis Cup title for Argentina in the fifth final it has played since 1981. Croatia won in 2005.
Going into the reverse singles Sunday, the Croats were leading 2-1 in the best-of-five-series after winning the doubles on Saturday.
"Delpo showed his huge heart after turning around the match that looked lost," said Argentina team captain Daniel Orsanic. "Federico played a perfect match under huge pressure. I'm really proud to be part of this fantastic team."
With the victory, Argentina became only the third team to come back from 2-1 down in the final since the World Group Davis Cup format was introduced in 1981.
Delbonis went up a break in the first two sets and added two more in the third when Karlovic sent a forehand wide for the victory.
The 41st-ranked Argentine passed the big-serving Croat veteran at ease throughout the 2:09-hour match.
"This is fabulous," Delbonis said. "We somewhat felt we could do it despite trailing after the doubles."
Karlovic, who returned to the Croatian team after a four-year absence, said he was "hugely disappointed."
"Maybe I'll feel better tomorrow, but tonight it is awful," he said.
The comeback started with Del Potro's heroics that included a finger injury in the fifth set against Cilic.
"I was calm, I hoped all the time that I could win and I did it," said Del Potro, displaying a bandage on his little finger on his non-racket left hand.
He said the injury happened in the fifth set while he was trying to catch a ball.
"I'm very happy to hand my team a chance of a victory," Del Potro said.
The match at Arena Zagreb that lasted nearly five hours saw plenty of drama created by two players with similar styles.
"It is very difficult to lose such an important match," Cilic said. "I feel a lot of disappointment."
Cilic had a furious start to race to a 3-0 lead including a break in Del Potro's first service game. The Croat capitalized on the first of three break points with a cracking forehand winner.
Del Potro fought back in the seventh game with Cilic under pressure at deuce. Two ground strokes, first a forehand long then a looping backhand, saw Del Potro break at the sixth attempt.
In the tiebreaker, Cilic raced to a 5-0 lead to seal the game with a slice backhand down the line that Del Potro could not control.
Under pressure after losing his fourth straight tiebreaker during the finals, Del Potro started the second set slowly and dropped his serve in the fifth game with a double fault and another in the seventh at love.
At that point, Cilic seemed to be cruising toward victory with some spectacular ground strokes that the tired-looking Argentine watched in disbelief.
But then Del Potro, who has recovered after two injury-hit years, showed the form he displayed in beating now top-ranked Andy Murray as Argentina knocked out holder Britain in the semifinals.
He broke Cilic in the 12th and 10th games in the third and fourth sets respectively to send the match into the decider.
"After losing the third set, I was still confident I could win as he looked tired," Cilic said. "I started pushing a bit too hard, handing him a chance for a comeback."
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. He served out the match to keep Argentina alive.