On a comeback trail after recovering from a shoulder injury, Kashyap dished out a gritty performance to stave off the challenge from World No. 15 Lee 21-16 10-21 21-19 in a match that lasted an hour and three minutes.
Fifth seed Sameer Verma, who also is making a comeback after recovering from a shoulder problem, made a positive start to his campaign by defeating Vietnam's Hoang Nam Nguyen 21-5 21-10 in another men's singles match.
World No. 59 Kashyap will face Hungary's Gergely Krausz, former National champion Sameer will take on Crotia's Zvonimir Durkinjak, Prannoy will meet Ireland's Joshua Magee and Dani will square off against Vietnam's Tien Minh Nguyen.
In women's singles, Rituparna Das defeated Canada's Rachel Honderich 21-16 21-18 and Sri Krishna Priya Kudaravalli beat USA's Maya Chen 21-13 21-16.
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However, it was curtains for Lakhanee Sarang, Abhishek Yelegar, Sai Uttejitha Rao Chukka, Reshma Karthik and Ruthvika Shivani Gadde.
Ruthvika also put up a gallant fight before losing 14-21 27-25 16-21 to Japan's Aya Ohori and Reshma's fight too ended with a heart-breaking 21-12 16-21 15-21 loss against Sofie Holmboe Dahl of Denmark in the opening round.
In men's doubles, Manu Attri and B Sumeeth Reddy defeated Canada's Jason Anthony Ho-Shue and Nyl Yakura 21-15 21-19, while Francis Alwin and Tarun Kona beat local combo of Yan Tuck Chan and Brian Chi Cheng 21-3 21-10.
Third seed Pranaav Jerry Chopra and N Sikki Reddy lost 21-19 19-21 17-21 to English pair of Ben Lane and Jessica Pugh in mixed doubles.
Women's doubles pair of Kuhoo Garg and Ningshi Block Hazarika lost 12-21 10-21 to Russia pair of Anastasia Chervyakova and Olga Morozova.
Fourth seeds Satwiksairaj Rankireddy and Chirag Shetty too suffered a 21-8 19-21 19-21 loss to Kohei Gondo and Tatsuya Watanabe of Japan.