Sweet potato decays easily due to its high respiration rate and reactive oxygen species (ROS) accumulation during postharvest storage. In this study, we explored the relationship between antioxidant capacity in leaves and storage properties in different sweet potato cultivars, the tuberous roots of 10 sweet potato cultivars were used as the experimental materials to analyze the storage property during storage at 11–15°C. According to the decay percentage after 290 days of storage, Xu 32 was defined as a storage-tolerant cultivar (rot percentage less than 25%); Xu 55-2, Z 15-1, Shangshu 19, Yushu, and Zhezi 3 as above-moderate storage-tolerant cultivars (rot percentage ranging from 25 to 50%); Sushu 16, Yanshu 5, and Hanzi as medium-storable cultivars (rot percentage 50–75%); and Yan 25 as a storage-sensitive cultivar (rot percentage greater than 75%). Meanwhile, analysis of the α-amylase activity in root tubers of the 10 sweet potato cultivars during storage indicated that α-amylase activity was lowest in the storage-tolerant cultivar Xu 32 and highest in the storage-sensitive cultivar Yan 25. Evaluation of antioxidant enzyme activities and ROS content in the leaves of these 10 cultivars demonstrated that cultivar Xu 32, which showed the best storage property, had higher antioxidant enzyme activity [superoxide dismutase (SOD), catalase (CAT), ascorbate peroxidase (APX), and peroxidase (POD)] but lower lipoxygenase (LOX) activity, hydrogen peroxide (H2O2) and malondialdehyde (MDA) contents, and superoxide anion radical (O2⋅–) production rates compared with those of the storage-sensitive cultivar Yan 25 and the medium-storability cultivars Hanzi, Yanshu 5, and Sushu 16. Additionally, principal component analysis (PCA) suggested that sweet potato cultivars with different storage properties were clustered separately. Correlation and heat map analysis further indicated that CAT, APX, POD, and SOD activities were negatively correlated with α-amylase activity, while LOX activity and MDA and H2O2 contents were negatively correlated with the storage property of sweet potato. Combined, our findings revealed that storage property is highly correlated with antioxidant capacity in sweet potato leaves and negatively correlated with α-amylase activity in tuberous roots, which provides a convenient means for the screening of storage-tolerant sweet potato cultivars.