Giorgi, a quarterfinalist at Wimbledon in 2018, used her aggressive game to claim a 7-6(5), 0-6 6-4 victory and set up a semifinal against Estonian Anett Kontaveit.
Sabalenka had won their only previous meeting and looked on course for a comeback victory when she reeled off the second set with a flurry of winners to level the match.
The Belarusian then earned the first break of serve in the deciding set but Giorgi's game re-ignited to claim victory.
"It was a great match and very good level," she said. "At the end I think it was more mental than physical."
Giorgi knocked out fifth seed Karolina Pliskova in the opening round, also in three sets, meaning it is the first time in her career that she has had two wins over top-10 players at the same tournament