Maze consolidated her lead in the overall standings after leading by over half a second after the first run and holding on to take top spot and 100 World Cup points ahead of Germany's Viktoria Rebensburg and Tessa Worley of France.
It was Maze's fourth World Cup victory since October and the 15th of her career.
In St Moritz she won Friday's opening super-combined, finished second behind Lindsey Vonn in Saturday's super-G and wrapped it all up with another
That almost matched Vonn's efforts of the previous week at Lake Louise, Canada when she swept the three races on the programme — two downhills and a
Gut, who had the third-fastest time on the second run, ended up 0.49 seconds behind Worley.
St. Moritz was the site of 21-year-old Gut's first World Cup victory in 2008 when she became the youngest female skier to win the Super G at the age of 17.
Switzerland's Dominique Gisin finished 10th in Sunday's competition.
Maze went fastest on the first run and was over half a second clear of Rebensburg and Worley and the threat from Vonn, who was fifth fastest, soon
vanished as the American champion made a bad mistake halfway down the piste to fall out of contention.
But Rebensburg, the reigning World Cup giant slalom champion, posted an impressive second leg time and Maze only just managed to hold onto her lead,
winning by a wafer-thin margin of 00.08 seconds.
For Worley it was her first podium placing since she won in Andorra in February.