The Slovenian, who shared the top of the podium with Swiss Dominique Gisin in the downhill, was quickest by 0.52 seconds from Jessica Lindell-Vikarby, with Italy's Nadia Fanchini 0.65 seconds back ahead of the second run.
Wet, murky weather, with snow falling at the top of the course, made conditions slippery but Maze took advantage of starting first to punch in a time that proved beyond the rest.
American teenager Mikaela Shiffrin made her Olympics debut and was in contention for a medal after setting the fifth-fastest time, less than a second behind Maze.
"I was nervous in the start gate but when I was in the gate I wasn't, I just wanted to ski," the 18-year-old told reporters at the rainy finish area.
Austria's super-G champion Anna Fenninger was fourth while Vancouver champion Viktoria Rebensburg was in touch in sixth.
Rebensburg's German team mate Maria Hoefl-Riesch failed to start because of a sore throat, meaning she only has one chance left, in Friday's slalom, to equal Janica Kostelic's Olympic record of four Alpine skiing gold medals.
Ninety skiers were on the start list for a race that traditionally includes skiers from many non-Alpine nations.
Among those making up the numbers was violin virtuoso Vanessa Mae representing Thailand. Racing under the name Vanessa Vanakorn, she finished 26.98 seconds behind the leader.
(Editing by Peter Rutherford)
