Stochastic homogenization is achieved for a class of elliptic and parabolic equations describing the lifetime, in large domains, of stationary diffusion processes in random environment which are small, statistically isotropic perturbations of Brownian motion in dimension at least three. Furthermore, the homogenization is shown to occur with an algebraic rate. Such processes were first considered in the continuous setting by Sznitman and Zeitouni [21], upon whose results the present work relies strongly, and more recently their smoothed exit distributions from large domains were shown to converge to those of a Brownian motion by the author [10]. This work shares in philosophy with [10], but requires substantially new methods in order to control the expectation of exit times which are generically unbounded in the microscopic scale due to the emergence of a singular drift in the asymptotic limit.