Conditional Monte Carlo (CMC) has been widely used for sensitivity estimation with discontinuous integrands as a standard simulation technique. A major limitation of using CMC in this context is that finding conditioning variables to ensure continuity and tractability of the resulting conditional expectation is often problem dependent and may be difficult. In this paper, we attempt to circumvent this difficulty by proposing a change-of-variables approach to CMC, leading to efficient sensitivity estimators under mild conditions that are satisfied by a wide class of discontinuous integrands. These estimators do not rely on the structure of the simulation models and are less problem dependent. The value of the proposed approach is exemplified through applications in sensitivity estimation for financial options and gradient estimation of chance-constrained optimization problems.