The 14th-ranked Italian, who won the doubles at Wimbledon last week, was broken three times in the opening set as she won just six points on her serve.
Appearing in her first WTA quarterfinal, the 147th-ranked qualifier from the United States dropped serve only once while Giorgi hit 11 double faults.
The fourth-seeded German, who won her first of three career titles here in 2009, broke Voegele's serve eight times in a match interrupted for 3 1/2 hours by rain at 1-1 in the second set.
Petkovic will next play Grace Min of the United States, who beat No. 8 Karolina Pliskova of the Czech Republic 6-2, 6-1 in just under an hour.
It's the first WTA semifinal for the 138th-ranked Min, who won two events on the lower-tier ITF circuit this season.