Hseih was down 2-5 in the last set, but saved a match point then pushed back with a five-game winning streak to seal the win.
The unseeded Hsieh earned her first appearance in the last 16 at Wimbledon, where she will face Dominika Cibulkova of Slovakia for a place in the quarter-finals.
After three lengthy multi-deuce games to begin the final set, all breaks, the World No.1 appeared to assert her authority with a pair of statement holds that moved her ahead, 5-2.
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The Canadian won seven of nine games on Day 2 of a third-round match suspended because of darkness the previous day, beating Austrian qualifier Dennis Novak 7-6 (5), 4-6, 7-5. 6-2.
His next round rival will be Jiri Vesely of Czech Republic. Raonic, the No. 13 seed from Thornhill, Ont., led 6-5 in the third set when play resumed. "She played really well".
Jelena Ostapenko (12), Latvia, def. Vitalia Diatchenko, Russia, 6-0, 6-4.
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Even at Wimbledon, the home of lawn tennis, updates of England's goals were spread by fans watching or listening to the match on their cell phones.
Hsieh, a former Wimbledon doubles champion, has never reached the quarter-finals of a Grand Slam.
Halep follows Caroline Wozniacki (2), Garbine Muguruza (3), Sloane Stephens (4), Elina Svitolina (5), Caroline Garcia (6), Petra Kvitova (8), Venus Williams (9) and Madison Keys (10) out of the tournament. "I had match point", Halep said.
Reigning French Open champion and world No.1, Simona Halep, joined the trend of top 10 seeds humbled out of the 2018 Wimbledon Championships. I think because I was exhausted, because I'm exhausted, I couldn't stay focused on every ball.
Van Uytvanck's reward is a date with No. 14 Daria Kasatkina, who stopped No. 17 Ashleigh Barty 7-5, 6-3.
With the momentum well and truly on her side, Hsieh broke again when Halep failed to return a big forehand and the Romanian was unable to rescue the situation. Also physically I feel exhausted.
"There's a shop in London very famous now, called Burger & Lobster", she said. She stayed there for every point.
The world number 48 will face Slovakia's Dominika Cibulkova, a 6-2 6-2 victor over 15th seed Elise Mertens, in the fourth round.