Defending champion Saina brushed aside Bulgaria's Stefani Stoeva 21-16 21-12 but world number 10 Sindhu failed to clear the first-round hurdle in the women's singles competition.
World championship bronze medallist, Sindhu played her heart out before losing 19-21 20-22 to Japan's Eriko Hirose in a 50-minute contest.
In the men's singles, Kashyap reached the second round after his opponent Daren Liew of Malaysia retired midway at 11-4.
However, women's singles player Arundhati Pantawane bowed out of the first round after suffering a 17-21 15-21 loss to top seed Xuerui Li of China.
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Men's doubles pair of Manu Attri and Sumeeth Reddy B lost 14-21 14-21 to German combo of Michael Fuchs and Johannes Schoettler in a 29-minute match.
Saina dished out a dominating game as she used her smashes and net play to prevail over her Bulgarian rival. The Indian opened up a 3-0 lead and then extended it to 11-6 before moving to 17-8. She pocketed the opening game.