This was only Kasatkina's second grand slam quarter-final, so she's still feeling her way in the game.
Goerges, seeded 13th, faces seven-time Wimbledon champion Serena Williams on Thursday for a place in Saturday's final.
Ostapenko claimed her first win over Dominika Cibulkova to book her spot in a first semi-final at the All England Cub without dropping a set.
French Open champion Ostapenko, 21, was clearly the more adventurous player in her encounter with unseeded Cibulkova, pumping 33 winners and 28 unforced errors.
Angelique Kerber reached her second Wimbledon final after nullifying the threat of Jelena Ostapenko.
Williams dropped her first set at this tournament as the 52nd-ranked Giorgi produced an impressive display of power and accuracy in the opener.
"I think there are no favourites any more", said Kerber of her next challenge. "When I feel like this is something that should not happen, I would maybe have sympathy with her".
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Court won the last of her slam titles at the 1973 US Open. This was her 42nd Grand Slam appearance to finally get through to the last four at a major tournament.
One player who has kept her composure as the mayhem has carried on around her has been Kerber.
The 29-year-old, who met up with Germany football Mats Hummels at Wimbledon on Wednesday, said: "I think Angie did a much better job than I did the last years because she was nearly in every semi-final".
Earlier, Kerber raced into her second Wimbledon final and fourth Grand Slam showpiece as the German crushed former French Open champion Jelena Ostapenko 6-3, 6-3 in 67 minutes.
"I feel good, I feel like I did better today, I had to", she told BBC Sport.
After the match, Serena revealed she was confident of overturning the deficit.
"I hope he's going to go even further in the tournament". "I still have a long way to go to be where I was".
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Shrieking and clenching her fists after every key point that went her way, Serena let out a gutteral "C'mon" once she wrapped up the set.
Meanwhile, former Wimbledon finalist Angelique Kerber stepped up her bid to return to the title match with a 6-3, 7-6 (7/5) win against Switzerland's Belinda Bencic. Dreams can quickly turn to nightmares on a tennis court, so the strategy meeting that Goerges had planned with her team for Wednesday was set to be crucial.
Kasatkina made an encouraging start, putting away a forehand victor to earn a break point in the first game, but Kerber escaped with a battling hold and the 14th seed gifted her a 2-0 lead after double-faulting twice in a nervy first service game.
"Sometimes I'm also complaining because I think it was a wrong call".
"It's my point", Cibulkova said, ignoring the fact she had stopped before Hseih's shot legally landed in.
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Greg Van Avermaet retains the yellow jersey and remains first in the general classification for the fourth consecutive stage. Bernard Hinault went on the attack, even in retirement, at defending champion, Chris Froome , over his salbutamol case .