We propose and analyze a Strang splitting method for a cubic semilinear Schrödinger equation with forcing and damping terms. The nonlinear part is solved analytically, whereas the linear part - space derivatives, damping and forcing - is approximated by the exponential trapezoidal rule. The necessary operator exponentials and Ø-functions can be computed effciently by fast Fourier transforms if space is discretized by spectral collocation. We show wellposedness of the problem and H4(T) regularity of the solution for initial data in H4(T) and suficiently smooth forcing. Under these regularity assumptions, we prove a first-order error bound in H1(T) and a second-order error bound in L2(T) on bounded time-intervals. Nonlinear Schrödinger equation; Strang splitting; error analysis; stability; wellposedness; regularity.