Stephens, who had lost her only previous match against Rybarikova, held serve throughout the second set for a convincing victory.
The American will next face Andrea Petkovic. The German led 5-2 in the second set against Yvonne Meusburger of Austria but was broken twice while serving for the match before wrapping up a 6-3, 7-5 victory.
Earlier, Camila Giorgi of Italy upset last year's finalist Julia Goerges of Germany 6-2, 6-3.
The 105th-ranked Giorgi, who won three qualifying matches to reach the main draw, avoided break points and converted the second match point on her serve.
"It was a good match," said the Italian, who broke Goerges three times. "I feel well on these hardcourts and I will try to keep going."
In the second round, Giorgi faces fifth-seeded Kirsten Flipkens of Belgium, who rallied to beat 17-year-old Donna Vekic of Croatia 2-6, 6-2, 6-2.
Austria's Patricia Mayr-Achleitner, who entered on a wild card, came back from a set down to defeat sixth-seeded Sorana Cirstea of Romania 1-6, 6-3, 6-4.
Also, seeded players Carla Suarez Navarro of Spain and Dominika Cibulkova of Slovakia reached the second round. No. 4 Suarez Navarro defeated Kristyna Pliskova of the Czech Republic 6-2, 6-3, and No. 7 Cibulkova came from behind to beat Annika Beck of Germany 3-6, 6-2, 6-1.
Cibulkova next plays Katarzyna Piter of Poland, who defeated Irina-Camelia Begu of Romania 6-2, 4-6, 6-3.