Kirchgasser moved up from third after the first run and skied the steep Fuente del Tesoro course in a two-run combined time of 2 minutes, 14.41 seconds.
Nicole Hosp, the Austrian who won the giant slalom at the worlds 11 days ago and leads the World Cup giant slalom standings, was second, 0.42 seconds behind. Tanja Poutiainen of Finland, who had led after the first run, was third.
The 21-year-old Kirchgasser finished fourth in the giant slalom at the worlds. She sprained her thumb in training Friday but said she took painkillers before the race and the injury did not affect her.
"You ski with your feet, not your thumb," she said.
Kirchgasser's previous best World Cup results were second in a super-combi in Reiteralm, Austria, in December, and third in a giant slalom at Aspen, Colo., in November.
She was the 2003 world junior slalom champion.
Anja Paerson, coming off a five-medal performance at her home worlds in Are, Sweden, finished fourth.