Hsieh, the world No. 48, battled back from 2-5 down in the final set and halted Halep, who served for the match at 5-3 in the decider.
Top-ranked Simona Halep has been knocked out by Hsieh Su-Wei of Taiwan at Wimbledon, losing 3-6, 6-4, 7-5 in the third round.
Victory for Hsieh means Karolina Pliskova is the only member of the tournament's top-10 seeds still in the draw at the All England Club.
It also further boosted Serena Williams's chances of becoming an eight-time Wimbledon champion.
On Monday, Hsieh will face 2014 Australian Open runner-up Dominika Cibulkova with a spot in the quarterfinals at stake.
When Halep led 4-1 and 5-2 in the final set it looked as though she would live to fight another day but Hsieh broke back and repeated the trick at 5-5 when she landed a sweetly-timed backhand into the corner and Halep could only net.
Djokovic was furious when he was denied a break of serve in the seventh game of the fourth set on Centre Court when the ball bounced twice before Edmund hit what was ruled a victor.
"A positive match against a player with a lot of energy, some tough moments", the world No.1 told the BBC.
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USA tariffs on steel and aluminum imports have provoked retaliatory measures against billions of dollars of American exports. US officials and business owners have warned Trump that the tariffs will be a major hit to American companies.
Blue Jays players looking forward to Bautista's return to Toronto
Marcus Stroman , one of the remaining healthy Toronto starters, took the mound Wednesday night, but he was far from peak form. The only difference was this time he was sporting an orange-brimmed New York Mets hat and an unfamiliar No. 11 on his sleeve.
Novak Djokovic slams Wimbledon crowd after being booed at All England Club
He was running towards the ball and he just hit it and he dropped his racket. I just think it's not necessary. "I've played in Serbia".
Nadal, with Spain out of the World Cup, was heading to watch England play Sweden. "I had match point", Halep said.
Three-time victor Novak Djokovic made the last 16 of a Slam for the 44th time with a 4-6, 6-3, 6-2, 6-4 win over Kyle Edmund, the last remaining Briton.
Later Saturday, three-time champion Novak Djokovic plays his first match this year on Centre Court against Britain's Kyle Edmund.
Nishikori, seeded 24, bids to make the last 16 for a third time and record his 350th career win.
For the first time in Wimbledon history, none of the top five women's seeds reached the round of 16.
Far less so in the men's bracket, although No. 4 seed Alexander Zverev did depart Saturday with a 7-6 (2), 4-6, 7-5, 6-3, 6-0 loss to Ernests Gulbis, a Latvian ranked 138th who is the first male qualifier since 2012 to reach Wimbledon's fourth round.
Alison Van Uytvanck backed up her impressive victory over Muguruza with an assured 6-2 6-3 success against 28th seed Annett Kontaveit.
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