Third seeded Dipika beat England's experienced Jenny Duncalf 11-8, 9-11, 11-6, 7-11, 13-11 in 75 minutes.
Seventh seeded Joshna upsett second-seeded Australian Rachael Grinham 11-1, 11-8, 11-6.
Rachael was taken to five games in the opening round, just hours after landing from Australia and was clearly feeling the effects.
"Playing Rachael is always hard," said Joshna. “I felt that she didn't play her usual game today. I am quite glad to have won in three."
Joshna will now face compatriot Dipika with whom she made history last year by winning India's first squash gold medal in the Commonwealth Games in the women's doubles in Glasgow.
"Really happy to win. Jen played really well today. I knew I had to stick in there and play the rallies out," said Dipika.
"When I was match ball (point) down, I told myself to play the rallies out and stay in there for as long as it takes. And that really helped."
Quarter-finals: 1-Amanda Sobhy (USA) bt Delia Arnold (MAS) 11-6, 11-3, 11-8 (25 minutes); 4-Sarah-Jane Perry (ENG) btALine Hansen (DEN) 11-5, 12-10, 9-11, 11-5 (45m); 3-Dipika Pallikal (IND) bt 5-Jenny Duncalf (ENG) 11-8, 9-11, 11-6, 7-11, 13-11 (75m); 7-Joshna Chinappa (IND) bt 2-Rachael Grinham (AUS) 11-1, 11-8, 11-6 (23m).