Cornet, coming off a win last month in Strasbourg, prevailed over fellow Frenchwoman Pauline Parmentier 6-3, 6-4 on the clay courts of the Nuremberg Tennis Club.
German crowd favorite Goerges, meanwhile, moved on to the second round with a 6-4, 7-5 defeat of Romanian Alexandra Cadantu in the inaugural edition of the WTA Tour stop in Nuremberg.
All of Monday's scheduled matches were washed out by rain and continuing inclement weather plagued the event Tuesday, resulting in more delays and postponements.
Top-seeded Jelena Jankovic saw her match suspended by darkness while holding a 6-4 lead over Arantxa Rus of the Netherlands.
Those managing to squeeze in a Round 1 win Tuesday included No. 5 Czech Lucie Safarova, No. 6 Spaniard Lourdes Dominguez Lino, No. 8 German Annika Beck, wild card Andrea Petkovic of Germany and American Julia Cohen.