Frida Hansdotter of Sweden finished second, a distant 0.86 seconds behind. Kathrin Zettel of Austria was third, 1.13 back.
American teenager Mikaela Shiffrin moved up from 13th after the opening run to finish sixth and maintained a 13-point lead ahead of Maze in the slalom standings.
The 17-year-old Shiffrin had won the past two slaloms and has three wins overall this season. In the opening run, she was fastest in the top section and second fastest in the last section, but U.S. coach Alex Hoedlmoser said she had three big mistakes in the middle.
Vonn finished 36th in the first run and failed to qualify for the second leg. Both runs were held in clear conditions, with the temperature below freezing.
In the overall standings, Maze has more than twice as many points as her nearest challenger, Maria Hoefl-Riesch, the Olympic slalom champion who went out in the first run.
Maze leads Hoefl-Riesch 1,654-806, and Vonn is third with 740 points.
Vonn won two slaloms four seasons ago, but the event has become her weakest discipline. She hadn't even entered a slalom since last season's World Cup finals.
It was Maze's seventh win of the season and first in slalom. She now needs only a downhill win to hold victories in all five disciplines this season.
Only two women have won in all five disciplines in a season Austria's Petra Kronberger in 1990-91 and Croatia's Janica Kostelic in 2005-06. The only man to do it was Marc Girardelli in 1988-89.
Maze's super-G win in St. Anton two weeks ago gave her wins in each discipline over her career.
"Tina is sensational this season," Zettel said. "She has shown her class here once again."
A special city event is scheduled for Moscow on Tuesday, the last race before the Feb. 5-17 world championships in Schladming, Austria.
Only two more traditional slaloms remain this season, although the parallel city event a mix between slalom and giant slalom also counts toward the slalom standings.
In Kitzbuehel, Austria, Marcel Hirscher won a World Cup slalom for the 10th time in his career, drawing roars from a home crowd that included Austria-born Arnold Schwarzenegger.
Hirscher became the first Austrian since Klaus Kroell in 2009 to win a race at the Hahnenkamm event, one of the classic stops on the men's World Cup calendar.
Ivica Kostelic of Croatia won the combined, which adds the slalom results to those from Saturday's downhill. He became the first skier to win that competition four straight years.
Hirscher was third in the opening leg but was fastest in the second run to finish in a combined time of 1 minute, 44.34 seconds for his 17th career title. It was Hirscher's first victory in Kitzbuehel.
"If you want to be among the greatest skiers, you have to win in Kitzbuehel at least once," Hirscher said. The defending overall champion extended his lead in both the overall and slalom World Cup standings.
In the overall title race, Hirscher has 1,035 points to lead Norway's Aksel Lund Svindal by 114.
Felix Neureuther of Germany was 0.77 back in second, and Kostelic finished third, 0.91 behind Hirscher. Ted Ligety of Park City was among those who failed to finish.
Ligety skied out after five gates, failing to score points for a second straight day after missing a gate in the downhill. He finished sixth in Friday's super-G and remained in third place in the overall standings with 736 points.
"Kitz weekend started well but went sideways fast. Literally and figuratively," Ligety wrote on Twitter.
In the combined, Frenchmen Alexis Pinturault and Thomas Mermillod Blondin were second and third. Kostelic matched Norway's Kjetil Andre Aamodt's record of six straight podium finishes in this event.
Hirscher, who was beaten by Neureuther in a slalom in Wengen, Switzerland, last week, has won five races and finished on the podium 13 times this season.
With each victory worth 100 points, the Austrian has now gained a stunning 700 points in eight slaloms this season. Neureuther trails him by 134 points in second.
"It's not getting easier, the pressure is immense," Hirscher said. "If you get so many podiums in one season, you expect yourself to finish top five in each race."
Hirscher used an attacking yet error-free second run to win the race after trailing Stefano Gross and Giuliano Razzoli in the opening run. Both Italians, however, had costly mistakes in their final runs and finished 11th and fourth, respectively.
"It was a very, very tough race," Neureuther said. "In the second run, I raced tactically smart. Marcel showed the aggression you need to win here. He was always going to be the man to beat."
Kostelic called himself fortunate after finishing on the podium despite being more than 0.4 seconds off the top three in his opening run.
"I was lucky today and I wasn't expecting a top-three finish anymore," said Kostelic, adding that he admired the run of Austrian veteran Benjamin Raich. "I was inspired by the way Benni was skiing today."
Raich posted the second-fastest time in the final leg, just 0.10 slower than Hirscher, to climb from 25th to fifth place.
In a difficult first run, 28 racers failed to finish. The slalom gates were placed by Italian technical coach Jacques Theolier after an initial course set by Croatia's Ante Kostelic, father of Ivica and former women's overall champion Janica, was rejected as being too difficult by the international ski federation.
According to men's race director Guenter Hujara, the course was "unskiable" and Kostelic refused to change it, prompting the jury to ask the Italian team to replace the gates.
The new course still proved challenging, with Ligety, Mario Matt of Austria, Manfred Moelgg of Italy, last year's winner Cristian Deville of Italy and slalom world champion Jean-Baptiste Grange of France unable to finish.
This is the last season Kitzbuehel will run the classic combined event of downhill and two slalom runs on consecutive days. The competition is likely to be replaced next year by a super-combined event with a speed race and one slalom run on the same day.
A limited group of slalom skiers will take part in a city event in Moscow on Tuesday, the last World Cup race before the Feb. 5-17 world championships in Schladming, Austria.
In Omaha, Neb., little-known Max Aaron won his first title at the U.S. Figure Skating Championships and helped knock down three-time champion Jeremy Abbott to the last step on the podium. With two quadruple jumps and an arsenal's worth of other tricks in his "West Side Story" program, Aaron finished with 255 points overall, almost four better than Ross Miner. Abbott, who had won three of the last four U.S. titles, dropped to third after a disappointing and flawed free skate.
Aaron was the U.S. junior champ two years ago, but has done little of note since then.