We show how the infinitesimal exchangeable pairs approach to Stein's method combines naturally with the theory of Markov semigroups. We present a multivariate normal approximation theorem for functions of a random variable invariant with respect to a Markov semigroup. This theorem provides a Wasserstein distance bound in terms of quantities related to the infinitesimal generator of the semigroup. As an application, we deduce a rate of convergence for Johansson's celebrated theorem on linear eigenvalue statistics of Gaussian random matrix ensembles.