We propose a novel ecohydrological model for the invasion of inland waters by the zebra mussel Dreissena polymorpha and test it against field data gathered within the Mississippi-Missouri river system in North America. This biological invasion poses major ecological and economic threats, especially due to the huge population densities reached by local zebra mussel colonies and the species' unparalleled dispersal abilities within fluvial systems. We focus on a quantitative evaluation, attempted here for the first time, of the individual roles and the mutual interactions of drivers and controls of the Mississippi-Missouri invasion. To this end, we use a multilayer network model accounting explicitly for zebra mussel demographic dynamics, hydrologic transport, and dispersal due to anthropic activities. By testing our results against observations, we show that hydrologic transport alone is not sufficient to explain the spread of the species at the basin scale. We also quantify the role played by commercial navigation in promoting the initial, selective colonization of the river system, and show how recreational boating may have determined the capillary penetration of the species into the water system. The role of post-establishment dispersal mechanisms and the effectiveness of possible prevention measures are also discussed in the context of model sensitivity and robustness to reparametrization.