Drought, one of the most common natural disasters that have the greatest impact on human social life, has been extremely challenging to accurately assess and predict. With global warming, it has become more important to make accurate drought predictions and assessments. In this study, based on climate model data provided by the Inter-Sectoral Impact Model Intercomparison Project (ISIMIP), we used the Palmer Drought Severity Index (PDSI) to analyze and project drought characteristics and their trends under two global warming scenarios—1.5 °C and 2.0 °C—in Central Asia. The results showed a marked decline in the PDSI in Central Asia under the influence of global warming, indicating that the drought situation in Central Asia would further worsen under both warming scenarios. Under the 1.5 °C warming scenario, the PDSI in Central Asia decreased first and then increased, and the change time was around 2080, while the PDSI values showed a continuous decline after 2025 in the 2.0 °C warming scenario. Under the two warming scenarios, the spatial characteristics of dry and wet areas in Central Asia are projected to change significantly in the future. In the 1.5 °C warming scenario, the frequency of drought and the proportion of arid areas in Central Asia were significantly higher than those under the 2.0 °C warming scenario. Using the Thornthwaite (TH) formula to calculate the PDSI produced an overestimation of drought, and the Penman–Monteith (PM) formula is therefore recommended to calculate the index.