Costazza finished the Hochstein course in a combined time of one minute, 54.97 seconds. Hosp was second, 0.68 seconds back. Finland's Tanja Poutiainen was third, 0.86 seconds behind the winner.
The victory makes the Italian the first non-Austrian to win a World Cup women's slalom race since December of last year, when Therese Borssen of Sweden won in Semmering, Austria.
American Lindsey Vonn, the overall World Cup points leader coming into Saturday's race, finished more than 3.5 seconds behind Costazza. Vonn fell to second overall behind Hosp, who has 514 points to the American's 480. Maria Riesch of Germany is third overall with 450 points.
Some information for this report was provided by AFP, AP and Reuters.