The dynamic compartmentalization of eukaryotic cells is a fascinating phenomenon that is not yet understood. A prominent example of this challenge is the Golgi apparatus, the central hub for protein sorting and lipid metabolism in the secretory pathway. Despite major advances in elucidating its molecular biology, the fundamental question of how the morphogenesis of this organelle is organized on a system level has remained elusive. Here, we have formulated a coarse-grained computational model that captures key features of the dynamic morphogenesis of a Golgi apparatus. In particular, our model relates the experimentally observed Golgi phenotypes, the typical turnover times, and the size and number of cisternae to three basic, experimentally accessible quantities: the rates for material influx from the endoplasmic reticulum, and the anterograde and retrograde transport rates. Based on these results, we propose which molecular factors should be mutated to alter the organelle's phenotype and dynamics.