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.
In the first set, Giorgi grasped the opportunity to go up 2-4 when Williams made some uncharacteristic errors on her service game and then consolidated the break to go up 5-2.
After a insane and awesome final game Angelique Kerber, the only player to reach at least the quarter-final stage in each of the Grand Slams this season, reached the seventh major semifinal as she beat Daria Kasatkina for the fourth time in her career in seven meetings. Up 5-4, Williams got to serve for the match, and she won the game easily.
Williams is the big favorite to win an eighth Wimbledon title with all of the top-10 seeded women already knocked out.
Rare is the player who can produce Williams' sort of pace on serves - Giorgi's average speeds were actually faster on first and second serves, and she delivered the match's first three aces - and mirror her power in groundstroke exchanges at the baseline.
Having taken the title in the last two years she played here, in 2015 and 2016, this quarter-final success takes Williams' winning streak at the All England Club to 19 matches.
Goerges' success is all the more surprising because she had lost in the first round for the last five years at Wimbledon and considered grass her weakest surface.
Julia Goerges, seeded 13, said she was not surprised that the top 10 women's seeds had exited before the quarter-finals.
Kerber is seeded 11th and, after a week of shock defeats for her title rivals, that makes her the highest ranked woman left in the draw.
She will meet German 11th seed Angelique Kerber for a place in Saturday's final.
Kerber, panting but calm, would finally serve it out, 7-5, but the young Russian threw down a marker: She will be back on this stage.
After more than a year out of the game, the former world number one warned she was "absolutely" returning to her brilliant best heading towards the pointy of the championships. Williams stepped up to the line at 5-4 and brought up match point with an ace before completing victory when Giorgi could only push a forehand into the net.
But the American's formidable record could not dampen Goerges' jubilation on Tuesday because as she admitted: "After five years of first-round losses (at Wimbledon), I have grown a lot".
Yes, no matter what the rankings or seedings say, she still looks as if she's capable of playing like someone who's No. 1. Just having two in the last four again is a boost to the country's tennis, Goerges said. I was not thinking too much about the match points.
But then Serena did what she said she does, becoming even greater because someone was playing her at their greatest.
Williams is 3-0 against Goerges, winning in straight sets each time.
However, injury cut that run short, rendering her unable to serve thanks to a pectoral muscle injury, and denying everyone a clash between Williams and Maria Sharapova.