We analyze the posterior contraction rates of parameters in Bayesian models via the Langevin diffusion process, in particular by controlling moments of the stochastic process and taking limits. Analogous to the non-asymptotic analysis of statistical M-estimators and stochastic optimization algorithms, our contraction rates depend on the structure of the population log-likelihood function, and stochastic perturbation bounds between the population and sample log-likelihood functions. Convergence rates are determined by a non-linear equation that relates the population-level structure to stochastic perturbation terms, along with a term characterizing the diffusive behavior. Based on this technique, we also prove non-asymptotic versions of a Bernstein-von-Mises guarantee for the posterior. We illustrate this general theory by deriving posterior convergence rates for various concrete examples, as well as approximate posterior distributions computed using Langevin sampling procedures.