For most practical purposes, the focus is often on obtaining statistical moments of the response of stochastically driven oscillators than on the determination of pathwise response histories. In the absence of analytical solutions of most nonlinear and higher-dimensional systems, Monte Carlo simulations with the aid of direct numerical integration remain the only viable route to estimate the statistical moments. Unfortunately, unlike the case of deterministic oscillators, available numerical integration schemes for stochastically driven oscillators have significantly poorer numerical accuracy. These schemes are generally derived through stochastic Taylor expansions and the limited accuracy results from difficulties in evaluating the multiple stochastic integrals. As a numerically superior and semi-analytic alternative, a weak linearization technique based on Girsanov transformation of probability measures is proposed for nonlinear oscillators driven by additive white-noise processes. The nonlinear part of the drift vector is appropriately decomposed and replaced, resulting in an exactly solvable linear system. The error in replacing the nonlinear terms is then corrected through the Radon-Nikodym derivative following a Girsanov transformation of probability measures. Since the Radon-Nikodym derivative is expressible in terms of a stochastic exponential of the linearized solution and computable with high accuracy, one can potentially achieve a remarkably high numerical accuracy. Although the Girsanov linearization method is applicable to a large class of oscillators, including those with nondifferentiable vector fields, the method is presently illustrated through applications to a few single- and multi-degree-of-freedom oscillators with polynomial nonlinearity.