Eighth seed Anand defeated Malaysia's Lim Chi Wing 10-12 11-6 11-7 12-10 to set up a clash with Vladimir Malkov, seeded second, while Rahul beat compatriot fourth seed Subhankar Dey 11-13 11-4 11-6 11-7 in another match.
Rahul will face England's Yuehang Wang next.
In women's singles, Vrushali beat local shuttler Anastasiia Sharapova 11-8 15-14 11-6 and will face third seeded Russian Natalia Perminova next.
Sixth seed Rasika staved off a challenge from Japan's Minami Kawashima 13-11 11-7 15-14 to set up a meeting with another Japanese Natsuki Nidaira, fourth seed.
Saurabh Sharma and Anoushka Parikh also reached the quarters after getting over Russia's Aleksandr Vasilkin and Anastasia Redkina 11-9 11-5 11-7 in 20 minutes. They will meet Malaysian combo of Chan Peng Soon and Cheah Yee See.