This paper presents a volume mesh finite element method (FEM) for linear advection-diffusion equations on two-dimensional surfaces embedded in \(\mathbb{R}^3\). The emphasis is put on the convection dominated regime and stabilization of the method by a surface variant of the streamline upwind Petrov-Galerkin (SUPG) technique. The main contribution of the paper is the presented error analysis. A nearly optimal error estimate in a mesh dependent norm is proved. The analysis is provided for the stationary problem only, but the numerical experiments show the accuracy of the method for both the evolutionary and the stationary case.