British No 2 Robson went out of her previous three tournaments in the first round, before managing to get her first win since the Australian Open.
The victory over Giorgi wasn't entirely straightforward as the match was decided in a final set, the result being 6-2 4-6 6-3.
The 19-year-old started well breaking three times in the opening set, but the Italian worked her way back to set up a final set.
Thankfully for Robson she was able to hold her nerve and eventually came out on top in a match with 14 breaks of serve.
Her second round tie sees her take on the French player Alize Cornet who received a bye in the first round.
Two former Wimbledon semi-finalists in Sabine Lisicki and Tsvetana Pironkova were both eliminated in the round one after losses to Simona Halep and Daniela Hantuchova respectively.
Svetlana Kuznetsova saw off Lourdes Dominges Lino with a 6-3 6-3 victory putting the two-time grand slam winner into the hat for the second round.