World No.1 Saina comfortably beat 67th-ranked Cheah Lyddia Yi Yu of Malaysia while World No.4 Srikanth pipped Hans-Kristian Vittinghus of Denmark in a tight match.
Defending champion Saina breezed past Cheah 21-12 21-10 to enter the last 16 of the competition. The second seed next plays Sun Yu of China.
World No.14 Sindhu, however, suffered a 21-18 15-21 23-25 loss against China's eighth seed Wang Yihan.
Srikanth beat Vittinghus 14-21 21-8 22-20 in the man's singles. He plays China's Tian Houwei on Thursday.
Kashyap went down against China's sixth seed Wang Zhengming, losing 26-24 18-21 20-22.
Top men's seed Chen Long beat RMV Gurusaidutt after a fight back to win 15-21 21-9 21-17.