With the favourite gone, the 34-year-old Brennsteiner built on his strong first-run lead and held his nerve on the second descent to edge Norway's Henrik Kristoffersen by 0.95 seconds.
The two-times Olympic medallist Kristoffersen had surged into the provisional lead with three skiers remaining, securing his 96th career podium.
Croatian Filip Zubcic produced the biggest charge of the afternoon, climbing 10 places to finish third.
"It means a lot for me," Brennsteiner, who won gold in the team event at the 2022 Beijing Olympics, said.
"The last years were quite good but I had tough times in my career and I'm 34 years now and my first World Cup win feels just amazing."
The victory also marked the Austrian's fifth World Cup podium in giant slalom.
Fresh off a super-G victory on Thursday and coming into the race after winning the giant slalom opener in Soelden last month, Odermatt was building a strong run when he lost control of his inside ski at a right-hand gate and slid out.
It was the Swiss's first giant slalom non-finish in almost a year, the last coming just down the road in Beaver Creek last December.
"A classic inside ski mistake, typical for this snow here. Too bad, that never really happens to me in training," Odermatt, who did not sustain serious injuries, told reporters.
His exit was one of several upsets in the first run, as Norway's Atle Lie McGrath, one of the season's early standout performers, also dropped out of contention.
