The Dane, who has topped the rankings for more than a year in total but has never won a grand slam title, fought off four match points before succumbing to the world number 136 in the opening match of the night session on Arthur Ashe Stadium.
Wozniacki's usual problem of struggling to dictate matches came back to haunt her and Giorgi was the aggressor throughout, hitting 46 winners to Wozniacki's 13.
The 23-year-old Wozniacki has come closest to winning her maiden grand slam title at the U.S. Open, reaching the final in 2009 and making semi-final appearances in 2010 and 2011.
But her lack of power against a big-hitting opponent yet again contributed to her demise.
She saved three match points at 2-5 in the third set, and another in the following game but Giorgi nailed a big forehand winner to end proceedings in two hours and 32 minutes.
Giorgi will meet fellow Italian Roberta Vinci in the fourth round for a place in the quarter-finals.
"It was amazing, this match, especially the first time here on this court," the 21-year-old said. "I never play with (Vinci) but for sure, it will be an interesting match. It will be fun and exciting."
(Reporting by Will Swanton, editing by Nick Mulvenney)
