Droughts and floods are common in tropical regions, including Rwanda, and are likely to be aggravated by climate change. Consequently, assessing the effects of climate change on hydrological systems has become critical. The goal of this study is to analyze the impact of climate change on the water balance in the Nyabugogo catchment by downscaling 10 global climate models (GCMs) from CMIP6 using the inverse distance weighting (IDW) method. To apply climate change signals under the Shared Socioeconomic Pathways (SSPs) (low and high emission) scenarios, the Soil and Water Assessment Tool (SWAT) model was used. For the baseline scenario, the period 1950–2014 was employed, whereas the periods 2020–2050 and 2050–2100 were used for future scenario analysis. The streamflow was projected to decrease by 7.2 and 3.49% under SSP126 in the 2020–2050 and 2050–2100 periods, respectively; under SSP585, it showed a 3.26% increase in 2020–2050 and a 4.53% decrease in 2050–2100. The average annual surface runoff was projected to decrease by 11.66 (4.40)% under SSP126 in the 2020–2050 (2050–2100) period, while an increase of 3.25% in 2020–2050 and a decline of 5.42% in 2050–2100 were expected under SSP585. Climate change is expected to have an impact on the components of the hydrological cycle (such as streamflow and surface runoff). This situation may, therefore, lead to an increase in water stress, calling for the integrated management of available water resources in order to match the increasing water demand in the study area. This study’s findings could be useful for the establishment of adaptation plans to climate change, managing water resources, and water engineering.