Los Angles, - Stosur, the sixth seed at the WTA Premier event, brushed past Jankovic 6-1 6-3 in just 74 minutes in Charleston on Wednesday (AEST).
The Australian 2010 champion was dominant, breaking serve five times to ease through and end a four-match losing streak against Jankovic.
It was a relatively good day for the seeds on the green clay.
Safarova, the Czech 15th seed and 2012 runner-up, was forced into a gruelling battle to beat Varvara Lepchenko 7-5 6-7 (6-8) 6-3.
Croatian 11th seed Lucic-Baroni was also pushed before overcoming Aleksandra Krunic 6-1 4-6 6-3.
Zhang Shuai did it easier, thrashing Misaki Doi 6-2 6-1, and Irina-Camelia Begu cruised past Cagla Buyukakcay 6-0 6-0.
The only two seeds to fall were Lauren Davis and Yulia Putintseva.
Davis went down to Maria Sakkari 7-5 6-4 and Putintseva was edged by Magda Linette 4-6 6-2 6-4.
Meanwhile, Shelby Rodgers battled past Veronica Cepede Royg 6-7 (1-7) 6-3 7-6 (8-6), Mona Barthel beat Jennifer Brady 7-5 6-4, Fanny Stollar won a battle between two qualifiers 6-3 7-5 against Asia Muhammad and Sara Errani crushed Grace Min 6-1 6-1.
Anastasia Rodionova, Annika Beck, Bethanie Mattek-Sands, Kristina Kucova and Daria Kasatkina also advanced. (QNA)
Source: Qatar News Agency