Noise-induced escape from a quasiattractor, and from a quasi-hyperbolic attractor with nonfractal boundaries, is investigated by means of analogue experiments and numerical simulations. It is found that there exists a most probable (optimal) escape trajectory, the prehistory of the escape being defined by the structure of the chaotic attractor. A general theoretical approach to the escape problem is described. The possibility of achieving analytic estimates of the escape probability within the framework of Hamiltonian formalism is demonstrated. For the quasiattractor, the optimal deterministic escape force is found.