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Wozniacki, who has been ever-present at the tournament since 2015, had lost her two previous matches in Auckland to the two-time defending champion Goerges, including in the final in 2018.
The Dane, however, showed no signs of fatigue from a two hour, 19 minute, three-set match against 2017 champion Lauren Davis that went late into Thursday's evening session to beat the 31-year-old Goerges 6-1 6-4 in just 75 minutes.
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"I have had some tough matches against Julia in the past and I knew this one would be tough as well," Wozniacki said.
"Even though it was 6-1 in the first set I still felt a lot of pressure because she has big serves and big shots so I needed to fight for every point."
Wozniacki will now face Jessica Pegula in the semi-finals after the American became the first player to qualify for the last four when France's Alize Cornet retired with a leg injury in the second set of their quarter-final.
Pegula had been leading 6-0 3-2 when Cornet retired.
Wozniacki's victory came about two hours after her Auckland doubles partner Williams overcame tricky winds to beat Germany's Laura Siegemund 6-4 6-3.
Top seed Williams will next meet either third seed Amanda Anisimova or Canada's Eugenie Bouchard.
Williams had little trouble against the 31-year-old German although the gusty breezes at Auckland Tennis Centre played havoc with her ball toss, especially when she was serving for the match at 5-3 in the second set.
She saved one break point in the final game but served an ace to get to match point then sealed it after an hour and 28 minutes when a Siegemund forehand return went wide while she was under pressure.
"She was an incredibly tricky player," Williams said. "The conditions definitely didn't help so ... it's good to get through that."
Wozniacki and Williams are also scheduled to play their doubles semi-final against the Belgian pair of Kirsten Flipkens and Alison Van Uytvanck in the evening session.
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