Rast's victory was her first in giant slalom.
"It feels incredible. But first of all, there was a tragic accident in my hometown this weekend, and I'm thinking of all the families affected. This weekend we race for them," Rast said after the victory.
At least 40 people have died and 119 others have been injured in an explosion and fire during New Year's celebrations in Crans-Montana, Switzerland.
Before the first run in Slovenia, Viaplay said during its broadcast that Sara Hector had a bad cold.
The reigning Olympic gold medalist in giant slalom didn't have a good first run and was sixth going into the second. Things went much better there, but there was still no new podium finish for Hector.
The 33-year-old veteran finished fourth, half a second behind winner Rast.
Julia Scheib of Austria and American Paula Moltzan were second and third, respectively. Moltzan was just three hundredths ahead of Hector.
Before the competition in Kranjska Gora, Hector took third place in Semmering, Austria, in the intervening days, and earlier in December she was second in the discipline in Tremblant, Canada.
Estelle Alphand, who Viaplay said was "a bit of a wreck," finished tied for 18th.
On Sunday, the Women's Alpine World Cup continues in Kranjska Gora with a slalom.
