Noise-induced escape from the basin of attraction of a quasi-hyperbolic chaotic attractor in the Lorenz system is considered. The investigation is carried out in terms of the theory of large fluctuations by experimentally analyzing the escape prehistory. The optimal escape trajectory is shown to be unique and determined by the saddle-point manifolds of the Lorenz system. We established that the escape process consists of three stages and that noise plays a fundamentally different role at each of these stages. The dynamics of fluctuational escape from a quasi-hyperbolic attractor is shown to differ fundamentally from the dynamics of escape from a nonhyperbolic attractor considered previously [1]. We discuss the possibility of analytically describing large noise-induced deviations from a quasi-hyperbolic chaotic attractor and outline the range of outstanding problems in this field.