London Olympics bronze medallist Saina staved off a spirited challenge from World No 44 Mette Poulsen of Denmark 21-19 23-21 in a 46-minute match.
World No 11 Saina, who won the bronze at the World Championship, will next face a tough opponent in eighth seeded Chinese Chen Yufei, who had clinched the other bronze at the Glasgow tournament in August.
In men's singles competition, Commonwealth Games champion Kashyap squandered an opening game advantage to go down narrowly 21-15 9-21 20-22 to Korean Lee Dong Keun in a match that lasted an hour and nine minutes.