Discontinuous unpredictable, asymptotically stable solutions of systems of quasilinear impulsive differential equations are studied in the article. The concept of a discontinuous unpredictable function has been extended to the class of functions of several variables, and the concept of unpredictability with respect to a time variable has been introduced. Examples with numerical simulations are presented to illustrate the theoretical results for discontinuous unpredictable solutions of quasilinear impulsive systems.