Changing climatic conditions have led to a significant increase in the magnitude and frequency of extreme rainfall events in the Central Andes of South America. These events are spatially extensive and often result in substantial natural hazards for population, economy and ecology. Here we develop a general framework to predict extreme events by introducing the concept of network divergence on directed networks derived from a non-linear synchronization measure. We apply our method to real-time satellite-derived rainfall data and predict more than 60% (90% during El Niño conditions) of rainfall events above the 99th percentile in the Central Andes. In addition to the societal benefits of predicting natural hazards, our study reveals a linkage between polar and tropical regimes as the responsible mechanism: the interplay of northward migrating frontal systems and a low-level wind channel from the western Amazon to the subtropics.