A two-dimensional (2D) finite-volume Eulerian-Lagrangian method (FVELM) on unstructured grid is proposed for solving the advection equation in free-surface scalar transport models. A backtracking band is defined along the backward trajectory of a side center as the dependence domain of a cell face, over which scalar concentration distribution is integrated to evaluate the advective flux through the cell face. Using the cell-face advective fluxes, a finite-volume cell update is finally carried out to obtain new cell concentrations, when mass is conserved both locally and globally by the unique flux at a cell face. The FVELM is then tested by a solid-body rotation experiment, a laboratory bend-flume experiment, and a real-river test (in a 365-km reach of the Yangtze River). In solid-body rotation tests, the FVELM is revealed to at least achieve a performance of existing second-order accuracy advection schemes. Relative to explicit Eulerian methods, the FVELM extends the dependence domain of a cell face from the upwind cell to the backtracking band, and therefore allows large time steps for which the Courant–Friedrichs–Lewy number (CFL) can be greater than 1. Accurate and stable FVELM simulations can be achieved at a CFL as large as 2–5 in these three tests. Efficiency issues of the FVELM are clarified by using the bend-flume test with refined grids (193,536 cells) on a computer with 16 cores; a parallel run using the FVELM is 14.2 times faster than a sequential run. In solving a transport problem (using 16 kinds of scalars and 16 cores), a parallel run using the FVELM is 2.3 times faster than a parallel run using an existing subcycling finite-volume method.