We propose the methodology of building the process-driven models for medium-term forecasting of spring floods (including catastrophic ones) in the mountainous areas, the hydrological analysis of which is usually much more complicated in contrast to plains. Our methodology is based on system analytical modeling of complex hydrological processes in 34 river basins of the Altai-Sayan mountain country. Consideration of 13 types of landscapes as autonomous hydrological subsystems influencing rivers’ runoff (1951–2020) allowed us to develop the universal predictive model for the most dangerous April monthly runoff (with ice motion), which is applicable to any river basin. The input factors of the model are the average monthly air temperature and monthly precipitation for the current autumn–winter period, as well as the data on the basin landscape structure and relief calculated by GIS tools. The established universal dependences of hydrological runoffs on meteorological factors are quite complex and formed under influence of solar radiation and physical–hydrological patterns of melting snow cover, moistening, freezing, and thawing of soils. The model shows the greatest sensitivity of April floods to the landscape composition of river basins (49% of common flood variance), then to autumn precipitation (9%), winter precipitation (3%), and finally, to winter air temperature (0.7%). When it is applied to individual river basins, the forecast quality is very good, with the Nesh–Sutcliffe coefficient NSE = 0.77. In terms of the accuracy of process-driven predictive hydrological models for the mountainous areas, the designed model demonstrates high-class performance.