The Azzurri have taken the wooden spoon at the past three Six Nations and slipped below Georgia in the world rankings, but they came out on top in their first meeting since 2003.
Tamaz Mchedlidze scored the game's first try after Tommaso Allan slotted through a ninth-minute penalty, but Michele Campagnaro crossed to put Italy, which fell to a huge 54-7 loss to Ireland last weekend, back in front.
Giorgi Tsutskiridze was sent to the sin bin in the 28th minute but it was not until Georgia were restored to their full complement that Allan kicked a second three-pointer and Mattia Bellini powered over in the corner.
Simone Ferrari extended Italy's lead three minutes after the restart and their advantage stood at 18 points when Allan dabbed down after Soso Matiashvili's penalty.
Tommaso Benvenuti's early tackle on Beka Bitsadze earned him a yellow card and gave the visitors a penalty try, but handling errors late in the match stopped them from forcing a tense finale.