In this work, we study the exponential stability of the stationary distribution of a McKean-Vlasov equation recently derived in \cite{de_masi_hydrodynamic_2015,fournier_toy_2016}. We complete the convergence result proved in \cite{fournier_toy_2016} using tools from dynamical systems theory. Our proof relies on two principal arguments. First, the linearized semigroup is positive which allows to precisely pinpoint the spectrum of the infinitesimal generator. Second, we use a time rescaling argument to transform the original quasilinear equation into another one for which the nonlinear flow is differentiable. Interestingly, this convergence result can be interpreted as the existence of a locally exponentially attracting center manifold for a hyperbolic equation.