Kirchgasser won the event with a combined-run time of 1 minute, 32.57 seconds and edged Veronika Zuzulova of Slovakia by 1.2 seconds.
Austrian Marlies Schild placed third but earned enough points to win the World Cup season championship in the women's slalom.
Vonn placed eighth Saturday but pushed her points total for the season to 1,980, which is 10 points better than the record that Janica Kostelic of Croatia set in 2006.
Vonn can capture the all-time point total of 2,000 held by Hermann Maier by finishing 12th place or higher in Sunday's giant slalom at Schladming.
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