An adjoint sensitivity method is presented for nonsmooth semiexplicit differential-algebraic equation systems (NDAEs) of index 1. The adjoint sensitivity system is derived for a generalization of Mayer-type functionals subject to NDAE constraints. Opposed to classical Mayer-type functionals that depend on the differential and algebraic variables evaluated only at the initial and final times, the generalization depends additionally on the evaluation at multiple interior time points. The derived adjoint sensitivity method extends and builds on the method for smooth DAEs by Cao et al. [SIAM J. Sci. Comput., 24 (2003), pp. 1076--1089] and on the method for nonsmooth ODEs by Ruban [J. Comput. Systems Sci. Internat., 36 (1997), pp. 536--542].