Goerges had won three of his previous four encounters with Wozniacki but could not get the better of the Dane this time despite serving seven aces.
"She's always given me a lot of trouble in the past, so I knew this wasn't going to be an easy one," Wozniacki said.
"I just kept fighting. I tried to be more consistent than her and I think I managed to do that."
The road ahead for Wozniacki does not get any easier. Her next opponent is Czech fourth seed Strycova, who clawed her way back from 5-2 in the second set to beat Coco Vandeweghe 6-4 7-5, winning 20 of the last 23 points.
Williams also had to dig herself out of trouble against Vesnina, who had won two of their three previous meetings.
The Russian led 4-2 in the second set before the 34-year-old American reeled off the last four games on the trot.
"Things seemed to be going on serve but then she had a big game for a break and another big game for a hold, and things were getting really interesting," Williams said.
"She was really picking up her level. But after that, I don't know, the games came and then it was over. That's what you have to expect every time. Nobody's going to give it to you."
Davis recovered after losing the opening set against Polish qualifier Urszula Radwanska to win 2-6 6-2 6-1.
(Reporting by Julian Linden; Editing by John O'Brien)
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