In the context of a heat kernel diffusion which admits a Gaussian type estimate with parameter β on a local Dirichlet space, we consider the log asymptotic behavior of the negative exponential moments of the Wiener sausage. We show that the log asymptotic behavior up to time t β V (x,t) is V (x,t), which is analogous to the Euclidean result.Here V (x,t) represents the mass of the ball of radius t about a point x of the local Dirichlet space. The proof uses a known coarse graining technique to obtain the upper asymptotic, but must be adapted for use without translation invariance in this setting. This result provides the first such asymptotics for several other contexts, including diffusions on complete Riemannian manifolds with non-negative Ricci curvature.