Maze made her World Cup debut at age 15 in Slovenia in January 1999, in a giant slalom race at Maribor. During her early World Cup seasons, Maze competed in slalom, giant slalom, and super-G, but scored World Cup points only in the first two disciplines. At the 2001 World Championships, Maze's best result was a 16th place in slalom. In January 2002, Maze won her first podium, a second place in giant slalom in Maribor. At the 2002 Winter Olympics in the United States, Maze finished twelfth in the giant slalom in her Olympic debut at Park City, Utah.