The American teenager trailed Maze by a massive 1.17 seconds after the first leg, and needed to finish ahead of the Slovenian to win the slalom crystal globe in her first full season on the circuit.
Shiffrin, who was fourth in the morning, had a tentative start to the second run but blazed down the bottom part of course for the fastest time of the afternoon.
When first-run leader Maze crossed the line in third, Shiffrin put her hands to her face and sank to her knees in the finish area.
"I didn't know I could make up all that time in one run," said Shiffrin, who also won the slalom world championship title last month. "I'm excited to reach my goals. It's amazing. I am still trying to find my best skiing but this was my best run of the season."
It was so good that only one skier, Bernadette Schild of Austria, was within one second of her time in the afternoon. Shiffrin finished in a combined time of one minute 55.60 seconds, with Schild 0.20 behind in second. Maze trailed by 0.35.
Maze had already clinched the overall World Cup title along with the giant slalom and super-G disciplines but was still visibly distraught at letting slip such a big lead to Shiffrin. While the American paraded her crystal globe in front of photographers, Maze stood with her head buried in her arm, sobbing.
"I didn't lose the slalom season just in this race," Maze told reporters from her native Slovenia. "I made a lot of mistakes and I had a chance to win it before Lenzerheide."
The 29-year-old Maze praised Shiffrin as a "really big talent, one of the biggest. She did a great job."
Shiffrin was quick to pay tribute to her rival, who set a new World Cup points record in one of the most dominating seasons the sport has seen.
"I actually want to thank Tina Maze," Shiffrin said. "She's probably going to punch me after this, but she's been very inspiring and helped me get to where I am. She's really my greatest idol this season."
Maze led Shiffrin by seven points in the slalom standings going into the race, and seemed to have wrapped up the title after taking such a big lead in the first run.
So how did Shiffrin steady her nerves to put down such an impressive run in the second?
"I didn't," she said. "I was freaking out. Oh my God, I was freaking out."
Shiffrin's mother Eileen was also nervous as she watched on, clapping enthusiastically as she leaned over a railing in a quiet VIP zone at the edge of the finish area.
"Un-be-lievable," Eileen Shiffrin told The Associated Press seconds after her daughter was declared champion. "She dug so deep in that second run. Oh my God, she finally skied the way she can ski."
Erin Mielzynski of Guelph, Ont., finished 10th, while Marie-Michelle Gagnon of Lac-Etchemin, Que., did no finish.
Shiffrin, who turned 18 Wednesday, is the first American World Cup slalom champion since Tamara McKinney in 1984.
With her victory, she went a long way to fulfilling predictions that she can follow teammate Lindsey Vonn as a superstar in Alpine racing and be one of the big profiles at next year's Sochi Olympics.
Vonn was 20 when she won her first World Cup race, 23 when she secured her first season-long World Cup title in any discipline, and 24 before winning her first major gold medal, at the 2009 world championships.
Shiffrin already has that trio of accolades three days after her 18th birthday. The Vail, Colorado, native is the youngest World Cup slalom title winner since Christa Zechmeister of West Germany 39 years ago.
After a breakthrough season, which finishes with a giant slalom on Sunday, Shiffrin will return to face the media spotlight back home in the U.S. on Monday after four months racing in Europe.
First up is an appearance on "Late Show with David Letterman" on Tuesday in New York City.
"Yeah, Letterman, I am so excited about that," said Shiffrin, who will graduate high school in the summer. "I hope I don't trip when I'm going up on stage."Suggest a correction