Halep and seventh seed Karolina Pliskova are the only two top 10 women still standing at Wimbledon, with the likes of Caroline Wozniacki, title holder Garbine Muguruza, former champions Venus Williams and Petra Kvitova and US Open victor Sloane Stephens all sent packing.
This is the first time that Hsieh, now ranked 48 in the world has reached the final 16 players in a singles tournament.
She will next face Belgium's Alison Van Uytvanck, who followed up her surprise win over defending champion Garbine Muguruza by beating Estonian Anett Kontaveit 6-2 6-3.
The 49th-seeded player from Chinese Taipei told the BBC, "Yes, this is the biggest win of my career and the first over a world No.1". Japan's Kei Nishikori celebrates winning his third round match against Australia's Nick Kyrgios.
Two-time champion Rafael Nadal reached the last 16 for the eighth time, making sure of keeping the world number one ranking.
East European 21-year-olds Jelena Ostapenko and Daria Kasatkina carried the flag for the Wimbledon seeds with straight-set victories on Court Three that took them through to the fourth round on Saturday.
Police have spoken to Novichok poisoning victim Charlie Rowley
Police officers guard a cordon in Salisbury, one of five locations in the city and Amesbury under investigation. Police are continuing to hunt for a contaminated container which they believe was handled by the pair.
However, Hsieh's unpredictable offence and never-say-die attitude paid dividends as she upset the world number one to secure a 3-6 6-4 7-5 victory. "I just want to feel free and enjoy the match", she added, about how she prepared herself for the tie.
Bencic, who staged a strong return to competitive play at Wimbledon after being sidelined for several months due to injury, will next face off against former world No. 1 Angelique Kerber of Germany, now world number 10.
The top seed fell to a stunning third-round defeat against world No.48 Hsieh Su-wei as the shocks kept coming in the women's singles.
Nadal has not been beyond the last 16 since reaching the final for the fifth time in 2011.
Djokovic next plays Russian world number 40 Karen Khachanov. By the time she played her third-round match at the All England Club, she was being watched with the type of concern normally reserved for endangered species.
"I just believe that I was not very positive on court".
Then there is Stephens, the US Open champion of previous year who struggled for form immediately after winning her home Slam, losing in the first round of the next four tournaments (including the Australian Open).
In that moment, Hsieh thought back to a fourth-round match against Lucie Safarova at Nottingham a year ago, which she lost in a third-set tiebreak.