Chinese No. 1 seed Zhang comfortably made it past Magdalena Frech of Poland in straight sets 7-5 6-3 to set up a clash with Eugenie Bouchard's conqueror Nao Hibino, while Hsieh fought back from a set down to eventually overcome Polona Hercog of Slovenia 4-6 6-1 6-3.
The duo, though, will not be joined in the second-round by third seed Yulia Putintseva, who was forced to retire hurt through injury after losing the first set of her match-up with Anna Karolina Schmiedlova.
Earlier in the day, home favourite Hibino came through 6-4 6-4 against Bouchard in another surprise result, continuing the former world No. 5's poor form in 2018.
Bouchard, who lost to Marketa Vondrousova in the second-round of the US Open last month, also fell at the first hurdle in the Oracle Challenger Series at the start of September against qualifier Kristie Ahn.