Basin-boundary crossing transitions, induced by large and sudden perturbations of a dynamical variable or of a control parameter, are reported in the bistable Belousov-Zhabotinsky reaction in a continuously stirred tank reactor. We provide a geometrical interpretation of these transitions by constructing a response diagram in (2 + 1)-dimensional (phase + parameter) space, that displays the geometry of the basin boundary together with the customary parameter dependence of steady states. The observed ease of basin-boundary crossing arises from the proximity of the basin boundary and the manifold of initial steady states. This is shown to be caused by the flow in phase space that is peculiar to stiff systems with fast and slow variables. On the mechanistic level, the elementary step is identified that accounts for the divergence of trajectories from the basin boundary.