The Italian had already beaten defending champion Karolina Pliskova en route to the last eight and pulled off another upset as she claimed a 7-6 (7-5) 0-6 6-4 win over Sabalenka.
Giorgi’s bid for the third WTA Tour title of her career will now continue with a semi-final against Anett Kontaveit on Friday.
The Estonian also needed three sets to book her place, bouncing back from losing the opener to beat Viktorija Golubic 2-6 7-6 (7-2) 7-5.
Daria Kasatkina was unable to build on her win over fourth seed Iga Swiatek in the last round as she missed out on a spot in the semi-finals.
The Russian took the first set against Jelena Ostapenko but failed to build on that success as the Latvian ran out a 1-6 7-5 6-2 winner.
In the final match of the day, Anastasija Sevastova was beaten by Elena Rybakina, who had earned her place in the quarters by beating second seed Elina Svitolina the day before.
World number 61 Sevastova, who eliminated Coco Gauff, won the first set 6-2, but Rybakina fought back to claim the next two 7-6 (9-7) 7-6 (7-5).
Rainy conditions ensured there was no play at this week’s other WTA event held in Bad Homburg, Germany, though one player was able to secure passage to the semi-finals.
Sara Sorribes Tormo claimed her place in the next round after Victoria Azarenka, her scheduled opponent, pulled out due to injury.
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