Watson produced a stunning performance to send sixth seed Anabel Medina Garrigues of Spain crashing out at the second-round stage with a 6-7 6-2 6-3 victory.
World number 71, Watson, is ranked 31 places lower than her opponent, but the 20-year-old from Guernsey managed to recover from losing the first set on the tie-break to come through in three.
Watson will next face Pauline Parmentier of France in the quarter-finals.
Robson, meanwhile, secured her place in the last eight with a routine straight-sets 6-4 6-4 win over Zhou Yi-Miao of China.
The 18-year-old British number one, ranked 56th in the world, beat qualifier Zhou with breaks in both sets to secure a relatively comfortable progression.
Robson will next take on Chang Kai-Chen of Chinese Taipei.
8-Laura Robson (Britain) beat Zhou Yimiao (China) 6-4 6-4
Heather Watson (Britain) beat 6-Anabel Medina Garrigues (Spain) 6-7(4) 6-2 6-3
7-Chanelle Scheepers (South Africa) beat Mirjana Lucic (Croatia) 6-2 6-2
Pauline Parmentier (France) beat 2-Zheng Jie (China) 6-3 6-4
- Anabel Medina Garrigues
- Pauline Parmentier