In this paper, a fourth-order compact finite difference method is proposed to solve the unsteady convection-diffusion equation. We first transform the convection-diffusion equation to a reaction-diffusion equation, which is then solved by a compact high-order method. The new method is unconditionally stable and fourth-order accurate in both temporal and spatial dimensions. It requires only a three-point stencil for a one-dimensional problem and a five-point stencil for a two-dimensional problem. Two numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.