Italy's Vinci, who won the event in 2010, got past Olga Govortsova of Belarus 6-0, 2-6, 6-0, while second-seeded Goerges of Germany moved on with a 6-2, 6-1 decision over Spain's Silvia Soler-Espinosa.
Meanwhile, Serbia's Jankovic won in a walkover when China's Peng Shuai retired in the first set due to a right shoulder injury.
Also logging a Day 2 victory at CK Sportcenter Kockelscheuer was American Venus Williams, who fired five aces in 6-3, 6-1 rout of Swiss wild card Belinda Bencic.
Elsewhere Tuesday, No. 5 seed Sabine Lisicki of Germany -- the Luxembourg runner-up in 2009 -- was upended by Belgian wild card Kirsten Flipkens 3-6, 6-3, 6-1, and No. 6 Sorana Cirstea of Romania dispatched Croatia's Petra Martic 6-2, 6-2.
Lourdes Dominguez Lino of Spain and German qualifier Annika Beck also logged triumphs Tuesday.