Noise-induced escape from a quasi-attractor, and from the Lorenz attractor with non-fractal boundaries, are compared through measurements of optimal paths. It has been found that, for both types of attractor, there exists a most probable (optimal) escape trajectory, the prehistory of the escape being defined by the structure of the chaotic attractor. For a quasi-attractor the escape process is realized via several steps, which include transitions between low-period saddle cycles co-existing in the system phase space. The prehistory of escape from the Lorenz attractor is defined by stable and unstable manifolds of the saddle center point, and the escape itself consists of crossing the saddle cycle surrounding one of the stable point-attractors.