The 23-year-old Cobolli, roared on by frenzied home support inside the SuperTennis Arena, clawed his way to a 6-3, 6-7(5), 7-6(15) victory with the third-set tiebreak one of the best ever seen in the competition's rich history.
Both players produced some spellbinding tennis in a series of punishing rallies in the breaker, with the big-serving Bergs looking as though he would extend the tie to the doubles.
But Cobolli, who also failed to convert his first six match points, would not be denied and eventually fired down an unreturnable serve to spark wild celebrations in the stands.
"In the five years I've been captain, I've never seen anything like it," said Italy captain Filippo Volandri. "This match was incredible; in the end, it was 5 per cent tactics and 95 per cent heart."
Earlier, Matteo Berrettini powered past Raphael Collignon 6-3, 6-4 to put Italy in front. It was Berrettini's 10th successive singles win in the competition.
Italy will try to complete the three-peat in Sunday's final against either Germany or Spain, who face off on Saturday.
Should they prevail, they would become the first nation to win three successive titles since the Challenge Round was abolished after the 1971 event.
Cobolli, who was a regular in the 2023 title-winning side, is playing as Italy's number one in the absence of world number two Jannik Sinner and Lorenzo Musetti and has risen to the challenge in magnificent fashion.
After thrashing Filip Misolic on Wednesday to seal Italy's comfortable quarter-final win over Austria, he was pushed much harder by the 43rd-ranked Bergs, who battled back from a slow start to stretch Cobolli to the limit.
"It's really tough to say something about this match," Cobolli said. "I played for all of my team, my family, and it's one of the best days of my life."
Italy, who are on a 12-tie winning streak, are the first nation to reach three successive Davis Cup finals since Australia in 2001, although Australia lost two of those.