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The 22 year-old was back at the scene of her stunning breakthrough Grand Slam title last year, where she beat Garbine Muguruza in three sets.
But it was a short stay on Margaret Court Arena with the 76th-ranked Giorgi retiring because of a thigh injury after Kenin clinched a tight first set 7-5.
"I hope she's all right ... but I'm glad to get through," Kenin said. "I was missing so many forehands but I finally made them to close out the first set."
The American was upset in the quarter-finals in the season-opening event in Abu Dhabi last month by Greece's Maria Sakkari, ranked 22.
Kenin reeled off three straight games to start the match before going off the boil as Giorgi went 4-3 ahead.
Kenin regrouped before Giorgi called for medical attention at the end of the set and retired.
The Moscow-born Kenin emerged as a serious force in last year's truncated circuit, also reaching the French Open final and rising to a career-high fourth in the world rankings.
She plays Germany's Mona Barthel or Jessica Pegula of the United States next.
World number one Ashleigh Barty and 2019 Australian Open champion Naomi Osaka are in action later on Tuesday.