A new technique is proposed for determining the response of multi-degree-of-freedom nonlinear systems with singular matrices and constraints subject to combined deterministic and non-stationary stochastic excitation. Decomposing the system response into a deterministic and a stochastic component leads to two subsystems of equations, namely, one subject to deterministic excitation and one subject to stochastic one. The latter is treated by resorting to a generalized stochastic linearization-based framework combined with state variable analysis. Specifically, a matrix differential equation governing the equivalent linear system under non-stationary stochastic excitation is derived and solved in conjunction with the deterministic set of equations governing the deterministic system response. The validity of the proposed technique is demonstrated by a pertinent numerical example.