The British number one, unseeded in China but one of the in-form youngsters in women’s tennis, broke the former US Open semi-finalist three times in the first set, and took an early lead in the second.
She was unable to close out the match as the world number 22 broke back at 5-1, but Robson held at the second time of asking to win in 92 minutes.
Robson, 18, got to the fourth round of the US Open last month to reach a career-high ranking of 74 and she looks set to improve on that after adding Zheng to the recent scalps of Kim Clijsters and Chinese number one Li Na.
She plays another Chinese, seventh seed Shuai Peng, for a place in the semi-finals.
Seeds Sorana Cirstea, Urszula Radwanska, Chanelle Scheepers and Alize Cornet also reached the quarter-finals of the tournament which saw top seed Marion Bartoli withdraw due to a stomach virus.