Alize Cornet was among a string of high-profile players to be knocked out on Tuesday and Giorgi, Annika Beck and Maria-Teresa Torro-Flor followed a day on.
World number 39 Giorgi fell at the first hurdle as Canadian qualifier Gabriela Dabrowski battled back to win 3-6 6-3 6-2 in just under two hours.
Italian Giorgi was made to pay for 13 double faults, despite saving 11 break points against the world number 236 from Canada.
Giorgi must have thought she was on course for a routine win when she broke in the opening game of the match before winning another two games on the Dabrowski serve.
Dabrowski had other ideas, though, as she broke to love in the third game of the second set. Giorgi was not given a look-in for the remainder of the set, which the 22-year-old from Ottawa wrapped up by claiming a second break.
Giorgi was rattled and Dabrowski took advantage, winning the opening five games of the decider before taking the first match point that came her way to claim an impressive win.
Beck's hopes of claiming her first title of the year were dashed by a 2-6 6-4 7-6 (7-2) second-round defeat by Lara Arruabarrena.
Arruabarrena was unable to take two break points that would have put her 3-2 up in the first set and Beck made her pay by claiming two breaks of her own to take the upper hand.
Only three games went with serve in the second set and a fourth break for Arruabarrena levelled the match, before the Spaniard came through a deciding tie-break.
Arruabarrena's compatriot Torro-Flor had a day to forget as the seventh seed was brushed aside 6-2 6-0 by Julia Goerges in the first round.
Jana Cepelova, Yulia Putintseva and Chanelle Scheepers progressed to the third round, while Silvia Soler-Espinosa and Mona Barthel avoided first-round exits.