Goerges also recovered from a set down to beat Kiki Bertens, 3-6, 7-5, 6-1.
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.
Or when, about 10 minutes later, she stretched for a lunging backhand victor to break at love and take control of the third set, then raised a fist, figuring a berth in her 11th semifinal at the All England Club was close at hand.
Williams faces German 13th seed Julia Gorges in Thursday's semi-finals.
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But, he was his natural self at the County Ground, content to play at his own pace, Rohit blinded everyone with his strokeplay. However, Pandya reaped the rewards of it in a game-changing second spell where he picked all of his wickets for just 16 runs.
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.
Thursday's (Friday NZT) other semifinal will also feature a German, two-time grand slam champion Angelique Kerber, against 2017 French Open victor Jelena Ostapenko of Latvia.
Goerges, who had never made it past the third round in her 10 previous appearances at the All England Club, advanced with a comeback victory over Bertens.
Williams took the set in convincing fashion, 6-3, and yelled her trademark, "Come on!" as she clinched the final point. "I'm just pleased with my game today".
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The 90th Wimbledon win of her career - the most among active female players - means Serena, now 181st in the WTA rankings, will return to the top 100 next week. "I think everyone here has a great chance so you have to take it", Goerges said.
Kerber took advantage of Kasatkina's erratic serving and errors to return to the last four.
Kerber is a former No. 1 who owns two Grand Slam titles and was the runner-up to Williams at Wimbledon two years ago.
The 21-year-old is the only female player yet to drop a set in this year's tournament. While it's obviously a monumental achievement and great moment for her career, Giorgi wasn't exactly in the friendliest of moods after the match. 'Each of us, seeded or not, I'm not seeded, we're just here to do our best.
The same could be said of both Roger Federer and Rafa Nadal, who, like Serena, continue to find new levels of quality despite their relative age in tennis.
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