The conditions in the Swiss Alps were challenging as intense fog and snow shortened the event, with the situation worsening as the race went on.
Curtoni clocked one minute and 09.40 seconds, 0.29 seconds ahead of Goggia. Switzerland's Corinne Suter took third place, 0.73 seconds behind Curtoni.
"With flat light, it's tough and you need to be brave. I think I was able to put all my braveness on the slope today and it went well," Curtoni said.
Goggia took second place but hit her left hand on the third control gate and broke two fingers. After the race she was transferred to Milan, where she was set to undergo surgery in an effort to be ready for the downhill race on Saturday.
Goggia is in fourth place in the overall standings, 100 points behind the leader, American Mikaela Shiffrin, who took sixth place on Friday.
Racing will continue throughout the weekend in St Moritz with a downhill on Saturday and a super-G on Sunday.