In this paper we study space discretizations of a general class of first- and second-order quasilinear wave-type problems. We present a rigorous error analysis based on a combination of inverse estimates with semigroup theory for nonautonomous linear Cauchy problems. Moreover, we provide refined results for the special case that the nonlinearities are local in space. As applications of these general results we derive novel error estimates for two prominent examples from nonlinear physics: the Westervelt equation and the Maxwell equations with Kerr nonlinearity. We conclude with a numerical example to illustrate our theoretical findings.