In this paper we consider the Dirichlet problem for a singularly perturbed elliptic equation of convection-diffusion type on a rectangular parallelepiped. For such a problem, $\eps$-uniformly convergent monotone difference schemes on {\it piecewise uniform\,} meshes are well known; their $\eps$-uniform order of accuracy does not exceed 1. Based on solutions of finite difference schemes on {\it piecewise uniform embedded\,} meshes by using the Richardson extrapolation technique, we construct a numerical solution that converges $\eps$-uniformly with the second (up to a logarithmic factor) order of accuracy. The given technique can be applied to the construction and justification of higher-order accurate numerical solutions for $n$-dimensional problems, where $n >3$.