In a men's singles contest that lasted for more than an-hour-and-a-half, Kashyap played out of his skin as he fought back to see off Hsu 15-21 21-18 21-19 at the Wuhan Sports Center Gymnasium.
Commonwealth Games champion, Kashyap will next take on seventh seed Chinese Zhengming Wang, who beat Thailand's Boonsak Ponsana 21-10 21-12 in another match.
The Indian has lost to Wang thrice, while defeated him once at the 2014 India Super Series.
The Commonwealth Games bronze medallist Sindhu, seeded eighth, will next meet Macao's U Teng Iok.
Indian men's doubles pair of Akshay Dewalkar and Pranaav Jerry Chopra failed to cross the second round hurdle as they lost 15-21 17-21 to fourth seeds Hiroyuki Endo and Kenichi Hayakawa of Japan.
Another men's doubles combination of Manu Attri and B Sumeeth Reddy, meanwhile, got a walk over to enter the third round.
Beaten during the India Super Series, Hsu had come well prepared in the match against Kashyap. His tactics did bother the Indian, who failed to get a hold of the match early on, allowing the Chinese Taipei shuttler to open up a 7-2 lead.
