Watson sealed her progression with an impressive performance against Spanish sixth seed Anabel Medina Garrigues, coming from a set down to seal a 6-7 (4/7) 6-2 6-3 win.
Next up for the world number 71 is a meeting with France's Pauline Parmentier, who also pulled off a second-round shock by beating second-seeded Zheng Jie of China 6-3 6-4.
Robson, the world number 60 and eighth seed in Osaka this week, sealed her place in the last eight with a 6-4 6-4 triumph over China's Zhou Yi-Miao.
Chang Kai-Chen of Chinese Taipei awaits Robson in the next round.
Also today, seventh seed Chanelle Scheepers of South Africa defeated Mirjana Lucic of Croatia 6-2 6-2.