The former world No 1 and tournament top seed was forced to dig deep against the big-serving German, winning 6-4 6-4 to book a last-four clash with Czech Barbora Zahlavova Strycova.
Wozniacki dropped her serve once and broke back twice during a tense first set, with Goerges saving five of the seven break points she faced.
The German kept up the pressure in the second set and after breaking Wozniacki a second time, the Dane showed her anger with herself as she swatted a ball away.
After she broke back, Wozniacki later slammed her racquet on the ground when losing game point at 4-3 and 40-15 before regaining her composure for the remainder of the match.
“She serves really big and has big shots, I tried to run every ball down and even though at points I didn’t feel comfortable I still just fought to the end,” said Wozniacki.
Third seed Venus Williams had to come from 2-4 down in the second set to beat Russian Elena Vesnina 6-2 6-4 in their quarter-final tie.
Williams cruised through the first set but was made to fight when the Russian broke her early in the second.
But as the Russian’s confidence grew, Williams lifted her performance to break twice in quick succession and take the match.
“Sometimes it’s hard to keep the focus when you’re winning, but whenever my opponent plays better I feel it raises my game,” said Williams, who will play fellow American Lauren Davis in the semis.
Zahlavova Strycova took nearly two hours to beat Vandeweghe, with the seventh seed American blowing a chance to take the game into a third set when up 5-3 in the second. She was broken serving for the set and then lost the next two games and the match.
Davis was taken to her third three-setter in as many games when she recovered from losing the first set to beat Polish qualifier Urszula Radwanska 2-6 6-2 6-1.