This paper aims to develop a robust two-dimensional coupled numerical model based on an unstructured mesh, which can simulate rapidly varying flows over an erodible bed involving wet–dry fronts that is a complex yet practically important problem. Using a modified spatial reconstruction based on the finite volume method, the well-balanced property is preserved, which is important for accurate and efficient simulation of morphological problems. In the present study, the central-upwind scheme is extended to simulation of bed erosion and sediment transport for the first time. It is demonstrated that the proposed scheme shows good accuracy and high efficiency. A modified shallow water system is adopted to improve the model. The shallow water equations, sediment transport equation and bed evolution equation are coupled in the governing system. Multiple test cases are employed to demonstrate the robustness, accuracy, and efficiency of the current model. Furthermore, with a field scale dam-break test case, the efficiency and accuracy of the central-upwind method is verified in comparison with other popular Riemann solvers. The effects of the additional source terms in the adopted modified shallow water equations are also investigated by comparing the numerical results with a laboratory study available in the literature. The proposed scheme can efficiently track wetting and drying interfaces while preserving stability in simulating the bed erosion near the wet-dry fronts. The added terms in shallow water equations can improve the accuracy of the simulation when intense sediment-exchange exists; the central-upwind method adopted in the current study shows great accuracy and efficiency compared with other popular solvers; the developed model is robust, efficient and accurate to deal with various challenging cases.