He will now play either top seeded Malaysian Lee Chong Wei or Simon Santoso of Indonesia. They are scheduled to play later in the day.
The 25-year-old from Hyderabad defeated Sri Lankan Niluka Karunaratne in a tough three-game battle at the Wembley Arena. Kashyap started off with a 21-14 win in the first game in 20 minutes but the Sri Lankan bounced back to take the second 21-15 in a time of 19 minutes. He won the decider 21-9. It took 21 minutes for Kashyap to take the final game.
In the second game, Kashayp fought hard by saving eight game points. Kashyap scored 13 points each on serve in the first and deciding games. Overall, the match lasted 66 minutes.
In the final game, Kashyap had a 12-point lead at one stage and he never allowed the Sri Lankan to come back.
He had earlier won two matches in the group stages to qualify for the round of 16, which was a knockout phase. Kashyap is coached by former India badminton great Pullela Gopichand.
Kashyap, in early 2000, was diagnosed with asthma and had feared his career was over. However, with medication he fought off asthma and continued to blossom as a player.
An elated Indian Sports Minister, Ajay Maken congratulated Kashyap. He tweeted (ajaymaken): "KASHYAP on his day is a World Champ! Congrats KASHYAP! Congrats Coach GOPI! We are proud of both of your wards Saina & KASHYAP!."
Gopichand had fallen in the round of 16 in 2000 Sydney Olympics, which was the previous best result for India in men's singles. Another Indian Deepankar Bhattacharya too had reached third round (16) in 1992 Barcelona Olympics when badminton made its debut at the Games.
Gopichand, who is coaching Indians now, was by court-side and had a quiet celebration with his pupil.
Kashyap trains at Gopichand Badminton Academy in Hyderabad.
In the preliminary stages, Kashyap, the world number 21, had beaten Tan Yuhan (21-14, 21-12) of Belgium and Vietnam's Nguyen Tien Minh (21-9, 21-14). He remained unbeaten in Group D with two wins. He topped the group and progressed to the knockout stage.
This is great news for Indian badminton after their players were knocked out in women's doubles (Jwala Gutta/Ashwini Ponnappa) and mixed doubles (Jwala/V Diju).
India's ace woman shuttler Saina Nehwal will play in the pre-quarterfinals.