Chasing her first back-to-back giant slalom triumph, Scheib found herself trailing at the halfway point of her second run before rallying to clock an overall time of one minute and 56.46 seconds.
Switzerland's Camille Rast finished second while Sweden's Sara Hector completed the podium, but not before a series of spills in the first run, where as many as 19 skiers did not finish.
"It's crazy, I never thought it would be the win. It was so tough, it was so bumpy, and I was so relieved when I came into the finish," said Scheib, who now leads Robinson by 88 points.
Robinson, who held the giant slalom leader's bib coming into the race, was among the first casualties when she tumbled during her opening run and failed to finish.
Hector had looked in command after the first run, leading with a time of 57.15 seconds -- just 0.02 seconds ahead of Scheib. Overall World Cup leader Mikaela Shiffrin could only manage eighth-fastest in the opening run.
The second run brought more drama when American Paula Moltzan, who appeared destined for a podium spot, crashed heavily on a turn and needed several moments to catch her breath before standing up and walking away.
The local crowd erupted when Scheib moved into top spot - 0.14 seconds ahead of Rast - and then held their collective breath until Hector completed her second run.
But the Swede could only manage third-fastest, leaving Scheib to celebrate on home snow.
"I want to continue like this, but I want to also focus a little bit on Super G. So we'll see," Scheib added.
Shiffrin, who finished sixth-fastest, still holds the overall lead with the American now sitting 114 points ahead of Robinson.