The American three-time overall champion was fourth after the opening run before an impressive closing section on her second try allowed her to take it with a combined time of 2 minutes, 24.43 seconds.
"I can't believe this. This is a big dream coming true," said Vonn, who was on a GS podium just once before. "A win in GS has always been so far out of reach for me. Last season, after finishing third in Spindleruv Mlyn, I knew that maybe one day I could win. But I definitely did not plan to do it here." Vonn edged Olympic champion Viktoria Rebensburg of Germany by 0.04 seconds, while Austria's Elisabeth Goergl was 0.40 behind in third.
Overall World Cup champion Maria Hoefl-Riesch had two disappointing runs to finish 3.13 seconds back in 24th. The German skier lost almost two seconds in her opening run and misjudged a turn while attacking in the second.
The discipline sweep feat has previously been achieved by Austria's Petra Kronberger, Sweden's Pernilla Wiberg, Croatia's Janica Kostelic and Sweden's Anja Paerson.
"I came into this race with not much confidence," said Vonn, who skipped most training sessions this week after hurting her hip in a crash during practice. "I just tried to have two good runs, to ski fast but no to do something special." Vonn also matched Paerson's career tally with her 42nd World Cup victory. Only Austria's Annemarie Moser-Proell (62), Switzerland's Vreni Schneider (55) and Austria's Renate Goetschl (46) have won more races.
Vonn admitted to nerves after she placed fourth in the opening run.
"I feared that I might hold back too much," said Vonn, who has won 35 of her races in downhill and super-G and has struggled for consistency in the technical races. "But my coaches told me before the second run to have confidence and to ski as fast I can." Vonn said offseason training had paid off but preferred not speculate on her chances of winning the overall title back from Hoefl-Riesch.
"Well, it's very early days," Vonn said. "I have a lot of momentum and confidence going into the rest of the season." Federica Brignone of Italy positioned herself for a first World Cup giant slalom by taking a commanding first leg lead, but she skied out early in her second run.
World GS champion Tina Maze of Slovenia, who many consider a contender for the overall title, could not live up to expectations and placed 23rd.
Rebensburg, who won here last year on her way to the discipline title, led Vonn by 0.26 after the opening leg but was still satisfied to take second.
"Reaching the podium in the first race is always a positive signal," Rebensburg said.
A two-time gold medalist in speed events at the worlds, Goergl said she was helped by completing several training runs on the glacier last week.
"I could prepare well for the conditions here and my material is OK. I have found a good balance," Goergl said. "It's a good start to the season on a hill I don't really like. I was surprised by the result but will still have some work to do for the next races." Tessa Worley of France, who won three GS races last season, lost her left pole after hitting a gate early in her first run but the worlds' GS bronze medalist skied on and finally took fourth, 0.80 behind Vonn.
In the first run, Finland's Tanja Poutiainen and Italy's Denise Karbon both crashed out after hooking a gate but were unhurt.
The season-opening World Cup weekend will conclude with a men's giant slalom on Sunday.
