Cibulkova is just nine places above Stosur in the world rankings, but the manner of the victory for the Australian was suprisingly convincing as she broke the Slovakian's serve six times to progress.
Meanwhile second seed Li Na powered into the second round of with a victory over Kirsten Flipkens.
Chinese Li was in control for the majority of the encounter with her Belgian opponent, as she sealed a 6-1 7-6 (9-7) victory.
The Australian Open champion needed just 32 minutes to take the first set, and kept her composure to win the second on a tie-break after Flipkens had rallied from a break down.
Third seed Agnieszka Radwanska won the battle of two beaten Australian Open semi-finalists, overcoming Canada's Eugenie Bouchard 7-6 (7-3) 6-2.
Simona Halep was the most emphatic winner of the day, though, the fourth seeded Romanian needing just one hour and two minutes to hammer German qualifier Julia Goerges 6-2 6-0.
Serbian world number seven Jelena Jankovic claimed a straightforward 6-4 6-3 victory against Ukraine's Elina Svitolina.
Former world number one Caroline Wozniacki saw off Ekaterina Makarova of Russia 6-2 6-2, but 2013 Wimbledon finalist Sabine Lisicki was made to work hard for a 6-2 6-7 (5-7) 6-3 success against Czech qualifier Karolina Pliskova.
Carla Suarez Navarro eliminated Yvonne Meusburger 7-6 (8-6) 6-4 in a match that was preceded by a centre-court tribute to forrmer British number one Elena Baltacha, who passed away after losing her battle with liver cancer aged 30 on Sunday.
Svetlana Kuzentsova is also through after beating Alize Cornet, while there were wins for Anastasia Pavlyuchenkova and Irina-Camelia Begu over Maria-Teresa Torro-Flor and Zhang Shuai respectively.