Hsieh saved the match point with a backhand victor before holding for 5-5 and breaking Halep in the next game.
The latest was fourth-seeded Alexander Zverev, who lost to the qualifier Ernests Gulbis, 7-6 (2), 4-6, 5-7, 6-3, 6-0.
And Hsieh's unusual game-plan matched her whimsical personality off the court.
Third seed Muguruza joined second seed Caroline Wozniacki, fourth seed Sloane Stephens, fifth seed Elina Svitolina, sixth seed Caroline Garcia and eighth seed Petra Kvitova on the way out, along with five-time Grand Slam champion Maria Sharapova. "I don't know how far I will go in this tournament, but I'm confident with my game at this time", the Argentine said.
"This is my list every year for my team". I enjoy everything here - the strawberries, the shops, everything.
But she did joke she takes advice from her six-year-old nephew on where to eat. Find us on Facebook too! McDonald's? Hmmm, my nephew is really picky.
The Taiwanese-Chinese duo saved two of four break points and converted six of 11, winning 60 of the 105 points contested to set up a second-round match against Christina McHale of the USA and Jelena Ostapenko of Latvia.
Once her post-match celebration meal is devoured, Hsieh can focus on extending her unexpected run at Wimbledon.
Japan's Kei Nishikori raced past 15th seeded Australian Kei Nishikori 6-1 7-6 (3) 6-4 to reach the Wimbledon last 16 for a third time in his career.
Halep is the first No. 1 seed to fail to reach the round of 16 at Wimbledon since Serena Williams in 2014.
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I also think just mixing in the takedown and just having that threat of the takedown, can shut down a fighter's offense. Miocic was not impressed and actually shoved the ref, but he was done and only taking more damage. "I'm excited.
But she insisted it was too soon to start dreaming of Grand Slam glory. I'm just realistic and honest.
"I was too exhausted. I'm very, very grateful and thankful for all this".
What will she do with the free time?
"But I always say the same".
"I enjoy a lot on the court more than before,"said Hsieh".
His win on Saturday was his third successive five-setter having also played three matches in qualifying.
"I had to run and fight for every point".
"I will have my chances, have to stay aggressive and try to dictate more to the rhythm of today rather than yesterday, that's for sure". "I try to practice normal with the girls, don't go too insane on the drop shot". "That's what I think".
"I'll go and watch some football -England, it's coming home or not?" he said in reference to a football song which is now being sung, hummed and whistled on nearly every street and in every pub. "I was pretty nervous at the beginning especially playing the first match of the day".