Vonn was fourth in the Super-G but lost control of a ski during an early turn in her slalom run and failed to finish.
Maze extended her lead in the overall standings by finishing in an aggregate time of 2 minutes, 1.76 seconds after the Super-G and slalom. She defeated Nicole Hosp of Austria by 0.88, while Kathrin Zettel of Austria was 1.08 behind in third.
Julia Mancuso of the United States was fifth after the Super-G, but slipped to 11th after posting the 23rd-fastest time in the slalom. World super-combined champion Anna Fenninger of Austria was seventh, 1.90 behind Maze.
The Slovenian finished runner-up to overall champion Vonn last season without winning a race. She now leads the American by 187 points despite Vonn's three straight wins last week at Lake Louise, Alberta.
"I am so strong because I trained hard for the season," said Maze, who has 14 World Cup victories. "I put in a lot of hard work. That makes me full of confidence."
Maze led by 0.01 after the Super-G ahead of Lara Gut of Switzerland, who dropped to fifth following the slalom.
"It's a nice course, but the visibility wasn't great," said Maze, who has won races in every discipline except for Super-G. "I am very satisfied, it feels good. I've had good Super-G runs in the past, but it never worked out in the end."
Maze tops the standings with 497 points. Maria Hoefl-Riesch of Germany, who missed out on the podium by 0.11 in fourth place, is second with 369. Next is Zettel with 320 and Vonn with 310.
Hosp, the 2007 overall champion, was seventh after the Super-G. She was fast enough in the slalom to earn her first podium finish since the race here last season.
"At ski racing, your mind plays an important role," said Hosp, who has not won a World Cup race since capturing a slalom in January 2008 at Maribor, Slovenia. "If you don't get the results it's hard to stay relaxed. I hope this race will help me for the rest of the winter."
This was the first of only two super-combined events this season. They count for the overall standings, but no crystal globe will be awarded to the discipline champion.
A regular Super-G will be held on the same course Saturday, followed by a giant slalom on Sunday.