The effect of applying lemon essential oil (LEO, Citrus limon L.) and oregano essential oil (OEO, Origanum vulgare L.), associated with modified atmosphere packaging (MAP), on the preservation of fresh cut grated anco squash (Cucurbita moschata) was studied. Anco squash obtained in Santiago del Estero, Argentina, was washed, sanitized, cut, peeled, and grated. Samples were subjected to seven different treatments: untreated control (TC), sodium hypochlorite at 150 ppm (TH), LEO (applied by spraying) 4 and 8 μL/mL (TL4 and TL8), OEO at 4 and 8 μL/mL (TO4 and TO8), and the combination TL4 + TO4. All samples were packaged in polypropylene trays covered with 35 μm polypropylene film to generate a passive modified atmosphere and stored at 5 °C for 5 days. Mesophilic aerobic bacteria, sensory quality (aroma and flavor), total phenolic content, and antioxidant capacity were evaluated throughout storage. Mesophilic counts increased in all treatments; however, samples treated with essential oils, particularly TL8, showed lower microbial growth than TC and TH at day 5. Sensory quality decreased over time, although TL4 and TL4 + TO4 maintained the highest flavor scores and better overall acceptance. Total phenolic content remained stable or increased slightly in essential oil-treated samples, with TO8 and TL4 + TO4 presenting the highest values at the end of storage. Antioxidant capacity was also enhanced by essential oils, and the treatment TL4 + TO4 maintained consistently high values throughout storage (0.29, 0.28, and 0.30 μg GAE/g TF on days 0, 2, and 5, respectively). According to the evaluated parameters, the combined use of lemon and oregano essential oils and MAP represents a promising strategy for preserving the microbiological, sensory, and functional quality of fresh-cut grated anco squash. In particular, the combination TL4 + TO4 showed the most balanced performance, maintaining sensory acceptance while enhancing antioxidant capacity and phenolic content. Further studies should evaluate different lemon–oregano oil ratios and their combination with other preservation technologies.