She'll be facing Serena Williams, the seven-time champion who hasn't lost a match at the All England Club since 2014 - though she missed last year's tournament while pregnant.
Angelique Kerber, seeded 11 and now the highest-ranked player left in the tournament, beat Belinda Bencic of Switzerland 6-3 7-6 (7/5) and will meet Daria Kasatkina, who ended Alison Van Uytvanck's run with a 6-7 (6/8) 6-3 6-2 win.
Kerber needed seven match points to close out No. 14 Daria Kasatkina of Russian Federation 6-3, 7-5 at Centre Court, while Ostapenko defeated 2014 Australian Open runner-up Dominika Cibulkova of Slovakia 7-5, 6-4 at a windy No. 1 Court.
And Drake looked proud as he turned up to support Serena Williams during her quarterfinal match at Wimbledon on Tuesday.
"I knew after the first set, "all right, let's go three sets".
Ultimately, Williams found a way to win 3-6, 6-3, 6-4, but Giorgi gave her all she could handle.
Williams was forced to withdraw from the fourth round of the French Open with a pectoral injury but has impressed in her bid to equal Margaret Court's record of 24 major triumphs.
Williams said her sister Venus - a five-time Wimbledon champion - had already offered to coach Olympia if she wants to play tennis.
Left-hander Kerber, runner-up to Williams at the All England Club in 2016, has been involved in six of the top 20 longest rallies here this year and says she has been trying to be more aggressive than her usually defensive style.
Serena Williams will play in her 35th Grand Slam semifinal against an opponent appearing in her first. She hit one at 122 miles per hour against Giorgi, delivered six of her seven aces in the final set, and won 44 of the last 54 points she served.
She will face German world number 10 Kerber on Saturday in a repeat of the 2016 showpiece won by Williams.
The United States president Donald Trump is going to visit Great Britain this weekend and he will attend the men's singles semi-final at Wimbledon on Friday if John Isner reaches the semi-finals.
Williams stayed strong on serve to see off Giorgi, despite the latter's own power. "'I'll just keep fighting", Williams said.
"I'm here just to prove that I'm back, and I feel like I'm back". "I still have a long way to go to be where I was".
With all the top 10 women's seeds now out, Williams is hot favourite to win an eighth title.
Having crashed out in the first round at Wimbledon for five consecutive times, Goerges' emotions spilled over at the end of the match after she embraced Bertens at the net.
Serena Williams may be on the verge of making history at Wimbledon, but the stress-free former champion insists she doesn't feel under any pressure.
With Angelique Kerber also in the last four, it is the first time two German woman have made the semi-finals of a Grand Slam since Steffi Graf and Anke Huber at the 1993 French Open.
"It was great matches", Giorgi said in her postmatch news conference.
"Wow, it's pretty fantastic", Goerges said. "She (Ostapenko) won a Grand Slam, as well".