Olympic bronze-medallist, Saina, who got a bye and a walkover in the first two rounds, was off the blocks with a 21-14 10-21 21-10 win over Japan's Nozomi Okuhara in a women's singles match that lasted for an hour and seven minutes.
The Indian will next take on fifth seed Tzu Ying Tai of Chinese Taipei.
Eighth seed Sindhu, a two-time bronze winner at the World Championships, set up a clash with top seed Li Xuerui of China in the quarterfinals after brushing aside Macau's Teng Iok U 21-8 21-9 in another women's singles match at the Wuhan Sports Center Gymnasium.
However, Commonwealth Games champion Kashyap, who had played a long match yesterday against Chinese Taipei's Jen Hao Hsu, could not continue his winning run as his gallant fight ended with a 23-21 17-21 8-21 loss to seventh seed Zhengming Wang of China.
Among other Indians, the men's doubles pair of Manu Attri and Sumeeth Reddy B sank 10-21 13 21 against third seeds Xiaolong Liu and Zihan Qiu of China.
The mixed doubles pair of Arun Vishnu and Aparna Balan also went down 13-21 5-21 to Kai Lu and Yaqiong Huang of China.
An All England finalist, Saina fought back from 3-5 and 4-6 in the first and third game to hand Okuhara her third defeat in as many encounters.

The girl from Hyderabad lagged initially but she soon opened up a 9-5 lead and even though Okuhara clawed back with a four-point burst, Saina was able to keep her nose ahead to grab the opening game.
Okuhara changed her tactics in the second game and surged ahead to 5-0 early on and then kept distancing herself from the Indian to roar back into the contest.
In the decider, Saina was back in her elements and after lagging 3-5, she drew parity to enter the break with a healthy 11-7 lead. Okuhara tried to break Saina's game but the Indian stamped her authority, reeling off nine straight points to leave the Japanese stranded.
In men's singles, Kashyap showed tremendous grit as he fought back from 0-5 down to grab an 11-8 lead at one stage and despite Wang clawing back at 12-12, the Indian kept breathing down his neck to eventually earn the early bragging rights.
However, Wang bounced back comfortably in the second game as he led 6-3 early on and then held a 11-9 advantage at the interval. The Chinese blasted five points on the trot to create a gap which Kashyap couldn't bridge.
Once the match went into the decider, it was always difficult as the fatigue of playing a gruelling prequarterfinal match showed on Kashyap's movement, allowing Wang to grab the advantage with both hands.

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