We consider the Ricci flow close to compact Einstein metrics. We study the relation between dynamical stability and instability properties of a given Einstein metric with respect to the Ricci flow and the local behaviour of the Einstein-Hilbert functional. We prove the following: Let $(M,g)$ is a compact Einstein manifold with Einstein constant $\mu$. If $g$ is a local maximum of the Yamabe invariant and if the lowest nonzero eigenvalue of the Laplacian satisfies $\lambda>2\mu$, then $(M,g)$ is dynamically stable, i.e. any Ricci flow starting close enough to $(M,g)$ exists for all time and converges to an Einstein metric close to $g$ as $t\to\infty$.