We extend the cell-centered finite element method (CCFE) [1] with imposed flux continuity and streamline upwind technique to solve the advection-diffusion equations with anisotropic and heterogeneous diffusivity and a convection-dominated regime on general meshes. The scheme is cell-centered in the sense that the solution is computed by cell unknowns of the primal mesh. From general meshes, the method is constructed by the dual meshes and their triangular submeshes. The scheme gives auxiliary edge unknowns interpolated by the multipoint fluid approximation technique to obtain the local continuity of numerical fluxes across the interfaces. In addition, the scheme uses piecewise linear functions combined with a streamline upwind technique on the dual submesh in order to stabilize the numerical solutions and eliminate the spurious oscillations. The coercivity, the strong and dual consistency, and the convergence properties of this method are shown in the rigorous theoretical framework. Numerical results are carried out to highlight accuracy and computational cost.