Semistability is the property whereby the solutions of a dynamical system converge to a Lyapunov stable equilibrium point determined by the system initial conditions. We extend the theory of semistability to a class of first-order evolution variational inequalities, and study the finite-time semistability. These results are Lyapunov-based and are obtained without any assumptions of sign definiteness on the Lyapunov function. Our results are supported by some examples from unilateral mechanics and electrical circuits involving nonsmooth elements such as Coulomb's friction forces and diodes.