Radwanska, seeded No. 1, defeated Greta Arn 6-2, 6-2 behind five service breaks. She won nearly 60 percent of the points on serve, including 80 percent when she was good on her first serve.
Goerges fought off seven of the eight break points she faced in a 6-3, 6-4 win over Anastasija Sevastova. She lost just six points on serve in the first set but 20 in the second. She suffered just one break, however.
Heather Watson advanced when fifth-seeded Sorana Cirstea retired from their match after Watson won the first set 6-3. No. 6-seeded Yaroslava Shvedova topped Lara Arruabarrena-Veci 6-3, 6-2 in another first-round match.
Also Tuesday, Marina Erakovic beat Stephanie Dubois 6-2, 6-1 and Romina Oprandi had little trouble in a 6-0, 6-2 victory over Nudnida Luangnam.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys