Vonn used a heroic second run to move up from third place and turn in a combined two-run time of 2 minutes, 22.20 seconds on the Gold Fox course in Maze's home-country Maribor.
Anna Fenniger of Austria was third followed by Germany's Maria Hoefl-Riesch.
Vonn said saw room for improvement after her first run and went all-out near the end of her second run. "I had not skied a giant slalom for a month," she said. "I was hoping for a good result but not to win but this course perfectly suits my style."
Maze, the overall women's World Cup points leader, admitted to a slight glitch near the top of her second run but was pleased to be on the podium. "It is an incredible feeling to race in front of so many people cheering for you! I could hear them from top to bottom," she said. "Of course it is not easy to race when everybody expects you to win."
Maze extended her lead in the point race. She increased her total to 1,554 points. Hoefl-Riesch remained a distant second with 748 points.