The former world No. 3 fell to Giorgi in straight sets 6-4, 6-2, during Day 2 at the Racquet Club of Memphis.
Third-seeded Lucie Hradecka of the Czech Republic had an easier go of it against Georgian Anna Tatishvili, dispatching her 6-3, 6-1 in other first-round action.
Also Monday, No. 4 seed Marian Erakovic of New Zealand rallied to overcome American Sloane Stephens 4-6, 6-0, 6-1, and Michaella Krajicek of the Netherlands toppled No. 6 Brit Elena Baltacha 6-2, 6-1.