A high-order Godunov-type scheme based on MUSCL variable extrapolation and slope limiters is presented for the resolution of 2D free-surface flow equations. In order to apply a finite volume technique of integration over body-fitted grids, the construction of an approximate Jacobian (Roe type) of the normal flux function is proposed. This procedure allows conservative upwind discretization of the equations for arbitrary cell shapes. The main advantage of the model stems from the adaptability of the grid to the geometry of the problem and the subsequent ability to produce correct results near the boundaries. Verification of the technique is made by comparison with analytical solutions and very good agreement is found. Three cases of rapidly varying two-dimensional flows are presented to show the efficiency and stability of this method, which contains no terms depending on adjustable parameters. It can be considered well suited for computation of rather complex free-surface two-dimensional problems.