Seventh seed Axelson defeated Kashyap 21-13, 21-19 in a match that lasted 43 minutes at the Haixia Olympics Sports Center to crash out of the $700,000 tournament.
Axelson, ranked World No. 6, dominated the first game completely and didn't allow his Indian opponent to take a lead. The second game, however, was much closer as Kashyap came out with a better performance and threatened to take the match into the decider. But Axelson held his nerves and eventually emerged victorious in the second game as well to move into the quarterfinals.
Kashyap had earlier moved into the second round after registering a 21-14, 21-13 win over World No. 21 Thammasin Sitthikom on Wednesday.
India's challenge in women's singles is already over after PV Sindhu and Saina Nehwal both lost their first round matches.
Sindhu lost to World No. 42 Pai Yu Po of Chinese Taipei 13-21, 21-18, 19-21 to Pai in a match while Saina lost to World No. 22 Yan Yan Cai 9-21, 12-21 to bow out of the competition. H.S. Prannoy had also crashed out after losing 17-21, 18-21 to Denmark's Rasmus Gemke in his opening round contest.
B Sai Praneeth is the only Indian left in the singles event as he will be facing Anders Antonsen from Denmark in his second round match later in the day.
( With inputs from IANS )