World No 59 Tanvi recovered from a game down to edge out the 18th ranked Gilmour 13-21 21-11 21-17 in a match that lasted for more than an hour.
She is the only Indian in women's singles as top shuttlers Saina Nehwal and P V Sindhu took a break after a hectic Uber Cup campaign.
Young Sourabh Varma also played out of his skin to see off Hong Kong's Nan Wei 19-21 21-14 22-20 and will next face the winner of the match between fourth seed Kenichi Tago of Japan and Korea's Dong Keun Lee.
In women's doubles Commonwealth Games gold-medallists Jwala Gutta and Ashwini Ponnappa defeated Japan's Ayane Kurihara and Naru Shinoya 21-18 21-16 to enter the second round.
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Jwala and Ashwini will now be up against sixth-seeded Koreans Kyung Eun Jung and Na Kim.
However, it was a shocking exit for top players Srikanth and Kashyap.
While Germany's Marc Zwiebler avenged his Thomas Cup loss against Srikanth with a 18-21 21-13 21-8 win, Kashyap was done in by world number 116 Riichi Takeshita 21-15 23-25 18-21 in a men's singles match that lasted for one hour and 18 minutes.
Later, RMV Gurusaidutt lost 10-21 21-15 13-21 to Japan's Takuma Ueda after a gruelling battle which lasted more than an hour.