The Polish world No. 28 was unable to provide an answer to an aggressive performance by Giorgi, ranked no. 100 and known for her hard-hitting but unpredictable game.
Last year Radwańska reached the final of the WTA Sydney International. Five years ago she won the tournament.
Radwańska on Wednesday saw off American qualifier Catherine Bellis 7-6 (4), 6-0.
A day earlier, the Pole beat Britain’s Johanna Konta 6-3, 7-5 in the first round of the Australian tournament.
Radwańska, a former world No. 2, is struggling to return to form after an uninspiring season last year. She ended 2017 ranked outside the Top 20 for the first time in a decade.