Sindhu, 19, held her nerves in the last six match points to upset World No.5 Yeon Ju Bae of South Korea 19-21 22-20 25-23 in 1 hour and 16 minutes at the Ballerup Super Arena.
The 11th-seeded Hyderabadi will now take on World No.2 Shixian Wang of China, who eased past Thi Trang Vu of Vietnam 21-12 21-7 in the pre-quarterfinals match.
Earlier Olympic bronze medallist Nehwal fought her way back from a game down to register a 14-21 21-18 21-12 victory over Sayaka Takahashi of Japan.
Saina will take on World No.1 Li Xuerui of China in the last-eight stage of the tournament.
In the opening game today, the Indian looked completely off colour and was never in contention. From the very beginning she started to trail behind and there was not even one instance in the game when Saina could catch her opponent.
The World No. 14 Japanese straight away opened up a 4-0 lead and throughout the game, Saina was lagging behind. The nearest she could reach her rival was at 5-6.
At one stage the Japanese had opened up a huge 13-8 lead and the Indian was seen playing only the catch-up game.
But just when it was looking as if it was all over for Saina, she maintained her clam to brought the gap down to 15-16 before catching Takahashi at 17-17.
At 18-18, Saina upped the ante and reeled off three straight points to take the match to the decider.
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