The dynamics of quasilinear systems with regular moments of impulses is investigated when a perturbation generated by an unpredictable sequence is applied. The existence, uniqueness, as well as asymptotic stability of unpredictable solutions are proved making benefit of a novel definition regarding regular discontinuity moments. To achieve the main result a Gronwall type inequality for piecewise continuous functions is utilized. The theoretical results are supported with an illustrative example. This is the first time in the literature that the existence of unpredictable solutions is demonstrated for impulsive systems comprising periodic terms and regular moments of impulses.