In this study, the unsteady convection-diffusion equation in one-dimension has been solved by using a hybrid matrix-collocation method which is based on Lerch and Taylor polynomials along with collocation points. The method reduces the solution of the given convection-diffusion equation with the initial and boundary conditions to the solution of a matrix equation corresponding to linear algebraic equations system with unknown Lerch coefficients. Also, the error estimation on technique related with residual functions is developed and some illustrative examples to show the effectiveness and convenience of the method are fulfilled. Moreover, the proposed algorithm can be used to solve other linear or nonlinear physical problems.