We propose two Euler–Maruyama (EM) type numerical schemes in order to approximate the invariant measure of a stochastic differential equation (SDE) driven by an α-stable Lévy process (1<α<2): an approximation scheme with the α-stable distributed noise and a further scheme with Pareto-distributed noise. Using a discrete version of Duhamel's principle and Bismut's formula in Malliavin calculus, we prove that the error bounds in Wasserstein-1 distance are in the order of η and [Formula presented], respectively, where ϵ∈(0,1) is arbitrary and η is the step size of the approximation schemes. For the Pareto-driven scheme, an explicit calculation for Ornstein–Uhlenbeck α-stable process shows that the rate [Formula presented] cannot be improved.