Goggia followed her downhill win at Bad Kleinkirchheim in Austria last week winning in 1 minute and 36.45 seconds, a margin of 0.47 seconds ahead of Vonn.
"It's a dream come true," said the 25-year-old after her fourth World Cup win and first in Italy, which comes weeks ahead of the Winter Olympics in South Korea.
"Lindsey made a mistake, but if she hadn't she would probably have won by a large margin. After my performance I wasn't that happy."
Vonn, with 11 wins in Cortina and the best time in both training runs this week, had been 0.14 seconds ahead of the Italian on the top half of the Olympia delle Tofane course.
But the 33-year-old American hit a bump and almost went into the netting, before recovering to finish on the podium.
"I still got second place which is outstanding," said Vonn. "Hopefully I won't get caught in a bump again tomorrow. It's always nice to have a second chance."
Friday's race was originally scheduled for Val d'Isere, France, last month but was moved because of poor weather conditions.
The second downhill takes place on Saturday with the super-G on Sunday on the slopes in the Italian Dolomites.
Overall World Cup leader Mikaela Shiffrin, competing for the first time in the downhill in Cortina, was third, 0.84 seconds behind the winner.