Endocytic vesicle formation is a complex process that couples sequential protein recruitment and lipid modifications with dramatic shape transformations of the plasma membrane. Although individual molecular players have been studied intensively, how they all fit into a coherent picture of endocytosis remains unclear. That is, how the proper temporal and spatial coordination of endocytic events is achieved and what drives vesicle scission are not known. Drawing upon detailed knowledge from experiments in yeast, we develop the first integrated mechanochemical model that quantitatively recapitulates the temporal and spatial progression of endocytic events leading to vesicle scission. The central idea is that membrane curvature is coupled to the accompanying biochemical reactions. This coupling ensures that the process is robust and culminates in an interfacial force that pinches off the vesicle. Calculated phase diagrams reproduce endocytic mutant phenotypes observed in experiments and predict unique testable endocytic phenotypes in yeast and mammalian cells. The combination of experiments and theory in this work suggest a unified mechanism for endocytic vesicle formation across eukaryotes.