High-resolution morphological modeling of fluvial processes with complex, rapidly varying flows has been limited so far by model accuracy or computational efficiency. One of the most widely used numerical algorithms is based on the total variation diminishing method, solved by either upwind or centered approaches. An upwind scheme preserves high accuracy but is complex and computationally demanding, whereas the simplicity and efficiency of a centered approach compromise the accuracy. The present paper extends a recent upwind-biased centered scheme originally developed for clear water and scalar transport over a rigid bed, to sediment-laden flows over an erodible bed. It does so by developing a fully coupled 2-D mathematical model using a finite volume method for structured grids. The complete set of noncapacity-based governing equations, involving the effects of bed deformation and sediment density variation, as well as the influences of turbulence and sediment diffusion, and the temporal and spatial scales needed for sediment adaptation, is solved at one time to obtain synchronous solutions for the entire computational domain. For stability, a two-stage splitting approach together with a second-order Runge-Kutta method is employed for the source terms. The model is verified in a number of tests covering a wide range of complex (sediment-laden) flows. The model is demonstrated to accurately simulate shock waves and reflection waves, but also rapid bed deformations at high sediment transport rates. The combination of high numerical accuracy and computational efficiency makes the model an important tool to forecast flood events in morphologically complex areas.