Mango fruits sourced from tropical yields have had a high commercial comeback from being viewed as susceptible to chilling injury under long storage durations. When the fruits are exposed to cold storage, this results in physiological changes due to the side effects of the storage on the fruits, expanding the rates of loss during the period between harvest and marketing. It is difficult to harvest mangoes as the fruits show varying maturities and are located in different positions on the trees. The purpose of this study was to test the idea that fruits’ location on the tree influences how the fruit behaves during cold storage. During two seasons (2019–2020), the impact of on-tree fruit location, i.e., sunny side (SUN; fruit exposed to the sun for most of the day), shade (SHA; fruit grown on the shady side of trees), and inside the canopy (INS; fruit grown inside the tree canopy), on the chilling sensitivity and the activities of antioxidant enzymes of ‘Zibda’ mangos stored at a low temperature (4 ± 1 °C) for 35 days was determined. In contrast to SHA and SUN mangos, INS fruits were shown to be progressively tolerant to low storage temperatures. These fruits also showed the highest activities of ascorbate peroxidase (APX), peroxidase (POD), catalase (CAT), and superoxide dismutase (SOD). In addition, the contents of O2− and H2O2 decreased in INS fruit during storage. Consequently, the cell membrane compartments were maintained, showing low accumulation of both malondialdehyde (MDA) and the protein carbonyl group (PCG) during storage. These results indicate that the fruit positions can also be considered at the time of harvesting for the classification of fruits before cold storage. This classification can also be added to the mango trading protocol to minimize the loss of economic returns by chilling injury.