In this article, we propose a second-order uniformly convergent numerical method for a singularly perturbed 2D parabolic convection–diffusion initial–boundary-value problem. First, we use a fractional-step method to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction, which gives a set of two 1D problems. Then, we use the classical finite difference scheme to discretize those 1D problems on a special mesh, which results almost first-order convergence, i.e., O(N−1+βlnN+Δt). To enhance the order of convergence to O(N−2+βln2N+Δt2), we use the Richardson extrapolation technique. In support of the theoretical results, numerical experiments are performed by employing the proposed technique. © 2017 Elsevier Ltd