Moser, ranked 725th in the world and given a wild-card pass into the tournament in her home country, was up 6-2, 4-3 when Barthel retired due to a shoulder injury. Moser broke Barthel's serve four times in eight games.
She moves into the quarterfinals against unseeded Andrea Hlavackova, who was a 5-7, 6-1, 6-0 winner over qualifier Viktorija Golubic. Hlavackova won nearly 70 percent of the points over the last two sets.
Also in the second round, eighth-seeded Karin Knapp got past Alexandra Cadantu 7-6 (7-5), 4-6, 6-1, recording three of her eight breaks in the final set.
Yvonne Meusburger completed the first round with a 6-0, 6-0 rout Wednesday of Jasmina Tinjic. Meusburger took 88 percent of the points on serving, dropping three points over six games.
Celebrity Families of 2014 [PHOTOS]