A new promising approach for visibility-sensitive path-planning problems is presented. The paper focuses on covert navigation where a mobile robot needs to plan a stealthy path to approach a designated destination in a cluttered environment. The aim is to minimize the robot's exposure to hostile sentries within the same environment. The approach can be adapted to work with different levels of initial knowledge the robot may have about both the environment map and the sentries' locations. The approach depends on estimating a cost value at each free-space location that presents the risk of being seen by any sentry. Based on the distance transform algorithm methodology, the minimum visibility–distance cost to a goal is calculated at each cell in the grid-based environment map. Moving along the steepest descent trajectory from any starting point generates an optimal covert path to a goal. The approach has been evaluated with both simulated and physical experiments. A number of test cases are presented. In each case, a path with considerable covertness, compared to a short path to the same destination, is generated. In addition to covert navigation, the approach is introduced briefly as a potential solution for other visibility-based path-planning problems.