Stephens, who has not played since the Fed Cup final in November, looked rusty against the big-hitting Giorgi and made 35 unforced errors as she lost 6-3 6-0 to the 26-year-old in just over an hour.
World number 13 Stephens has not won a match since beating Madison Keys in the final at Flushing Meadows in September.
Two-time champion Azarenka pulls out of Australian Open
She lost both singles encounters in the Fed Cup against Belarus, lost twice in the WTA Elite Trophy event in Wuhai and suffered first-round losses in two tournaments in China.
Giorgi will next meet twice Wimbledon champion Petra Kvitova, who went through when last year's surprise Australian Open semi-finalist Mirjana Lucic-Baroni retired after losing the first set 6-1.
"I trained in Italy, my home, and felt so very good. I did six weeks of preparation. It was great. Very good," said Giorgi, who missed the back end of last season with an elbow injury.
"I made a few errors at the start but then I played my game and it was good after that."
Another former Slam champion also progressed with Germany's 2016 Melbourne Park title winner Angelique Kerber saving two match points to outlast Lucie Safarova 6-7(3) 7-6 (8) 6-2.
Kerber's victory sends her into a second round clash with multiple grand slam champion Venus Williams.
Slovakian dynamo Dominika Cibulkova also reached the last 16 with a quickfire 6-3 6-1 demolition of eighth seeded Latvian Anastasija Sevastova.
Sydney's weather continued to wreak havoc with Sunday's extreme heat replaced by storms with hail, lightning and rain all halting the Kerber-Safarova clash.
The Australian Open, the opening grand slam of the year, runs from January 15-28.
Australian Open 2018: All you need to know about the first Slam of the year