Grazing bifurcations occur in nonsmooth dynamical systems when the behavior of the system changes discontinuously associated with a grazing contact. By a grazing contact, the interaction of an attractor with a discontinuity of the system is meant. In mechanics, such events represent transitions between non-impacting and impacting dynamics. In this paper, conditions are found for the persistence of a local attractor in the vicinity of a grazing trajectory. While previous papers dealt with periodic grazing trajectories, the authors consider quasiperiodic trajectories. In analogy to the periodic case, the catastrophic loss of a local attractor is associated with the repeated application of a square-root term that appears to lowest order in the normal-form expansion. The main tool employed in this paper is the discontinuity-mapping approach, where a Poincaré mapping for the nonsmooth system is obtained as a composition of a smooth Poincaré mapping and a discontinuity mapping. The results of the paper are illustrated by several model examples.