The prevailing view of intra-Golgi transport is cisternal progression, which has a key prediction--that newly arrived cargo exhibits a lag or transit time before exiting the Golgi. Instead, we find that cargo molecules exit at an exponential rate proportional to their total Golgi abundance with no lag. Incoming cargo molecules rapidly mix with those already in the system and exit from partitioned domains with no cargo privileged for export based on its time of entry into the system. Given these results, we constructed a new model of intra-Golgi transport that involves rapid partitioning of enzymes and transmembrane cargo between two lipid phases combined with relatively rapid exchange among cisternae. Simulation and experimental testing of this rapid partitioning model reproduced all the key characteristics of the Golgi apparatus, including polarized lipid and protein gradients, exponential cargo export kinetics, and cargo waves.