Qualifier Giorgi, 22, broke Sharapova eight times at Indian Wells, including every game in the opening set, to win her maiden match against the Russian.
In an incredible first set, Sharapova won just 12 points on serve as the opening six service games went the way of the receiver.
When Giorgi finally broke that trend in the seventh game she claimed a 4-3 lead before breaking Sharapova again and then clinching the first set by holding once more.
Sharapova took control of the second set in the sixth game, breaking Giorgi to lead 4-2 and despite dropping serve in the ninth game, the world number five levelled the match at a set apiece in the next game.
In the final set, Giorgi broke Sharapova three more times, including in the 26-year-old's final two service games, as the world number 79 triumphed in two hours and 37 minutes.
Sharapova (fourth) was one of six seeded players to lose on Monday with only five advancing to the fourth round.
Stephens overcame Ivanovic 7-6 (7-3) 6-4 despite trailing 3-1 in the second set.
The 17th seed from the United States held on over the next three games, saving four break points as Ivanovic looked to go 5-2 ahead, before breaking the Serbian 11th seed to level at 4-4.
Stephens then held to love and the broke Ivanovic again, converting her fourth match point to win in one hour and 44 minutes.
In a great day for the underdogs, four of the six seeded players beaten were eliminated by lower-ranked opponents.
Anastasia Pavlyuchenkova (21st seed) and Samantha Stosur (16th) joined Sharapova and Ivanovic in that group.
Stosur went down to Flavia Pennetta (20th) 6-4 3-6 6-1, while Aleksandra Wozniak defeated Pavlyuchenkova 6-1 6-7 (7-9) 6-0.
The other seeds to bow out were Svetlana Kuznetsova (27th), who lost 6-3 2-6 6-0 to Petra Kvitova (eighth), and Ekaterina Makarova (23rd), who was undone by Dominika Cibulkova (12th) 6-4 7-5.
In other results, top seed Li Na defeated Karolina Pliskova 6-3 6-4 and Alisa Kleybanova won 6-1 6-3 over Maria-Teresa Torro-Flor.