Climate change is known to cause alterations in weather patterns and disturb the natural equilibrium. Changes in climatic conditions lead to increased environmental stress on embankments, which can result in slope failures. Due to wetting–drying cycles, expansive clayey soil often swells and shrinks, and matric suction is a major factor that controls the behavior. Increased temperature accelerates soil evaporation and drying, which can cause desiccation cracks, while precipitation can rapidly reduce soil shear strength. Desiccated slopes on embankments built with such soils can cause surficial slope failures after intense precipitation. This study used slope stability analysis to quantify how climate-change-induced extreme weather affects embankments. Historic extreme climatic events were used as a baseline to estimate future extremes. CMIP6 provided historical and future climatic data for the study area. An embankment was numerically modeled to evaluate the effect on slope stability due to the precipitation change induced by climate change. Coupled hydro-mechanical finite element analyses used a two-dimensional transient unsaturated seepage model and a limit equilibrium slope stability model. The study found that extreme climatic interactions like precipitation and temperature due to climate change may reduce embankment slope safety. The reduction in the stability of the embankment due to increased precipitation resulting from different greenhouse gas emission scenarios was investigated. The use of unsaturated soil strength and variation of permeability with suction, along with the phase transition of these earthen embankments from near-dry to near-saturated, shows how unsaturated soil mechanics and the hydro-mechanical model can identify climate change issues on critical geotechnical infrastructure.