The victory, in a time of one minute and 46.28 seconds with bib number three, was the 30-year-old Val Gardena skier's first on the World Cup circuit and her first podium since she was third in Saalbach in March 2024.
Germany's Kira Weidle-Winkelmann was sixth out of the start hut and took second place, 0.20 slower than Delago.
Vonn, racing to third from 11th, was a further 0.06 behind on a Di Prampero piste shrouded in thick fog on the middle section.
Delago's younger sister Nadia, a downhill bronze medallist at the 2022 Olympics, finished 10th with the pair embracing at the finish.
"I was so nervous this morning, and I couldn't wait to go down to ski this slope. I really like this course," said Delago, who was also the fastest in Friday's training.
Vonn, in astonishing form as she prepares for an Olympic comeback at next month's Milano Cortina Games at the age of 41, stretched her lead in the downhill standings to 144 points over Germany's Emma Aicher.
The American now has two wins, a second place and two thirds from the five downhills this season, while Delago has six podiums from her entire career.
Vonn won a World Cup super-G in Tarvisio in 2011 but said she could not recall much about the mountain.
"Everyone said 'oh, it's a big advantage because you're the only one that's skied here'," she said.
"It's not an advantage because I don't remember it.
"But I do remember the fans, and I do remember the atmosphere.
"The snow was really soft, it was really hard to see in the middle, and I made a pretty big mistake, but I'm happy to be on the podium again.
"For me, what's important is to be consistent, and I'm really happy that I could do that despite the conditions."
There is one more downhill to come, in the Swiss resort of Crans-Montana on January 30, before the women head for the Games in Cortina d'Ampezzo.
Tarvisio also hosts a women's super-G on Sunday.