Williams looked to be in a spot of bother when she surprisingly lost the opening set 6-3 and struggled with consistency from the back of the court.
The 36-year-old is returning to Wimbledon this year after missing the tennis tournament last summer due to her pregnancy.
Two successive aces, clocked at 103 and 106 miles per hour, gave the 21-year-old the first set and the 2017 French Open champion tightened her grip on the contest when she ended another flurry of breaks to go 4-2 up in the second.
"Every match starts from zero", Goerges said of her semifinal matchup against Williams.
"After the first set, I said "okay let's go three sets", I just kept fighting".
"It's definitely family before tennis for me", she said to Harper's Bazaar UK. I was just trying to stay focused on my serve.
Ultimately, Williams found a way to win, 3-6, 6-3, 6-4, but Giorgi gave her all she could handle.
"For me, there's so much farther I want to go to get back where I was, and hopefully go beyond that".
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Going a break down elicited a typically angry response from Williams, who surged into a 40-0 lead on Giorgi's serve in the subsequent game.
Del Potro will hope his second trip to the Wimbledon quarter-finals mimics his first run into the last eight.
"I feel like I'm getting to where I want to be".
"At the French Open I had all that pressure, now it's gone", former junior Wimbledon champion Ostapenko said.
You could understand the 13th seed's bewilderment, after she came from a set down against Kiki Bertens to win 3-6, 7-6, 6-1.
Rapper friend Drake and fellow singer Justin Timberlake were among those rooting for her on Centre Court.
With all the top seeds sent home, the Wimbledon favorite is now Serena Williams.
"I hate losing. That's no secret", the 23-times major victor said after digging deep to deny Camila Giorgi a famous victory with a rousing 3-6 6-3 6-4 quarter-final comeback win over the unseeded Italian.
In the men's draw, Juan Martin del Potro completed a hard-fought fourth-round win over Gilles Simon 7-6 (7-1) 7-6 (7-5) 5-7 7-6 (7-5).