This work explored the use of natural deep eutectic solvents (NADES) for extracting phenolic compounds with antioxidant activity from Uruguayan olive pomace (OP), a by-product of olive oil mills. Among nine NADES tested, lactic acid-glucose (La-Gc) was the most effective solvent for phenolic compound extraction. Further investigation focused on optimizing the extraction process using La-Gc. Response surface methodology was employed to analyze the impact of extraction temperature, water content in NADES and solid–liquid ratio on total phenols content (TPC) and antioxidant activity (FRAP and DPPH assays). Temperature and solid–liquid ratio greatly influenced TPC and antioxidant activity, while water content in NADES only showed significant influence on antioxidant activity. The optimum extraction conditions that maximized TPC and antioxidant activity were 80 ◦C, 68% (w/w) of water in NADES and solid–liquid ratio of 0.014 g/mL, showing a TPC of 15.56 mg GAE/g db, FRAP of 178.14 µmol FSE/g db and DPPH of 72.75 µmol TRE/g db, with hydroxytyrosol content of 1.24 mg/g db. These values were significantly higher than those of extracts obtained from conventional solvents under identical extraction conditions. These results highlight the suitability of the present extraction method using La-Gc as solvent to obtain phenolic compounds with antioxidant activity from OP.