Having shaken off the rust following her lengthy lay-off after the birth of her daughter Olympia in September, Williams is on the verge of a 10th Wimbledon final appearance.
In the opposite half of the draw sits Serena Williams, who is seeded No 25 despite being ranked 183rd in the world.
All 10 of the top women's seeds suffered shock defeats in the opening four rounds for the first time in the Open Era.
Ostapenko, the 12th seed, overpowered Slovak Dominika Cibulkova 7-5 6-4 to reach Wimbledon's final four for the first time, and without dropping a set.
"I was maybe the better player in the first set but didn't actually take the chances I would have loved to take, but still found a way to somehow come back and close out that match".
"I don't follow tennis", the Italian said when the subject of Williams' return came up before today's quarter-final.
Preview: Serena Williams is excited to face an opponent whose game she's familiar with in the Wimbledon quarterfinals.
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Williams won Wimbledon on her last two visits in 2015 and 2016 before missing last year's event due to her pregnancy. "I mean, I'm hoping this is, like, a new thing", she said.
Watched from the stands by her friend and mega-star rapper Drake, Serena struggled to hit the right notes in the first set.
On Wednesday, the 29-year-old will be playing in his first Wimbledon quarter-final since 2013, when he went on to lose to Novak Djokovic in an epic semi-final.
Once Williams had figured out how to break the Giorgi serve, though, in the fourth game of the second set, the momentum changed.
"Every match starts from zero", Goerges said of her semifinal matchup against Williams.
Williams is 3-0 against Goerges, winning in straight sets each time. But this is only my fourth tournament back (so) I don't feel pressure, I don't feel like I have to win this.
Williams faces Italy's Camila Giorgi in the quarterfinals at 7 a.m. Tuesday at the All England Club.
Goerges also battled back from a set down to beat Dutchwoman Kiki Bertens 3-6 7-5 6-1 in the last of the women's quarterfinals. Argentinean del Potro was up two sets to one when bad light stopped play on Monday evening.