Assuming the heat kernel on a doubling Dirichlet metric measure space has a sub-Gaussian bound, we prove an asymptotically sharp spectral upper bound on the survival probability of the associated diffusion process. As a consequence, we can show that the supremum of the mean exit time over all starting points is finite if and only if the bottom of the spectrum is positive. Among several applications, we show that the spectral upper bound on the survival probability implies a bound for the Hot Spots constant for Riemannian manifolds. Our results apply to interesting geometric settings including sub-Riemannian manifolds and fractals.