With Halep out, none from the top 5 seeds would make it to the round of 16, the first time ever in the Wimbledon history.
Meanwhile on Centre Court, three-time champion Novak Djokovic battled back from a set down to defeat Britain's Kyle Edmund 4-6, 6-3, 6-2, 6-4, despite a double-bounce controversy.
Ostapenko eventually ran out a comfortable 6-0 6-4 victor.
Gilles Simon, France, def. Matthew Ebden, Australia, 6-1, 6-7 (3), 6-3, 7-6 (2).
Big-hitting Canadian Milos Raonic, the 2016 finalist, has reached the fourth round after beating Austrian Dennis Novak 7-6 (5) 4-6 7-5 6-2.
De Minaur valiantly saved two match points before Nadal confirmed victory with a short, sharp volley.
Jelena Ostapenko (12), Latvia, def. Vitalia Diatchenko, Russia, 6-0, 6-4.
Alison Van Uytvanck, Belgium, def. Anett Kontaveit (28), Estonia, 6-2, 6-3.
Russian's 14th seed Daria Kasatkina roared back from 4-1 down in the opening set to beat Australian Ashleigh Barty 7-5 6-3 in the third round.
Kerber will next meet Switzerland's Belinda Bencic, who ousted 27th seed Carla Suarez Navarro earlier Saturday.
"For me, I just try to enjoy the tennis, enjoy the city, enjoy the food". She stayed there for every point. I think because I was exhausted, because I'm exhausted, I couldn't stay focused on every ball. "I try to practice normal with the girls, don't go too insane on the drop shot". "My muscles are gone".
Victory would see him reach the last 16 of a major for the 44th time to take him second on the all-time list. Of the top 10 seeds, only seventh-seeded Karolina Pliskova has survived into the second week and the Czech is already in deep water, having never previously gone beyond the second round here. "Because every day, something unusual is happening in the draw".
Past year she knocked seventh seed Johanna Konta out of the French Open.
Halep battled with her own game as well as the crafty spins and tactics of her 48th-ranked opponent but still looked like emerging as the victor when she led 5-2 in the third set. I just want to feel free and enjoy the match.
(It's worth noting that the diminutive Hsieh served at an average speed of 82 miles per hour.) At the realization she'd won the match, Hsieh let go a scream and then a steady stream of tears.
Hsieh had 26 winners against 36 unforced errors.
Hsieh saved the match point with a backhand victor before holding for 5-5 and breaking Halep in the next game. 'I will not find the excuses about this match, she deserved to win, but still, I'm sad about myself today, ' Halep said. She did a lot of that during her news conference, making jokes and cracking herself up.
At Wimbledon, she is still far from her best, but has been cruising through the opposition.
Two-time champion Rafael Nadal reached the last 16 for the eighth time and made sure of keeping the world number one ranking.
On Monday, Hsieh will face 2014 Australian Open runner-up Dominika Cibulkova with a spot in the quarterfinals at stake.
Halep will be gone, off for some much-needed rest and relaxation.