Consider the following stochastic differential equation (SDE) dX_t=b(t,X_t−)dt+dL_t,X₀=x,driven by a d-dimensional Lévy process (L_t)_t≥0. We establish conditions on the Lévy process and the drift coefficient b such that the Euler–Maruyama approximation converges strongly to a solution of the SDE with an explicitly given rate. The convergence rate depends on the regularity of b and the behaviour of the Lévy measure at the origin. As a by-product of the proof, we obtain that the SDE has a pathwise unique solution. Our result covers many important examples of Lévy processes, e.g. isotropic stable, relativistic stable, tempered stable and layered stable.