The recently developed Wiener Path Integral (WPI) technique for determining the joint response probability density function of stochastically excited nonlinear systems is extended herein to account for systems with singular diffusion matrices. Among others, indicative examples include partially forced systems, hysteresis modeling via additional auxiliary state equations, as well as certain electromechanical energy harvesting systems. Specifically, the governing equations of motion can be represented as an underdetermined set of stochastic differential equations (SDEs), coupled with a set of deterministic ordinary differential equations (ODEs) acting as constraints. Next, appropriately defining the Lagrangian function of the system leads to a constrained variational problem to be solved for the most probable path, and thus, for the system response PDF. Two numerical examples pertaining to both linear and nonlinear constraint equations are considered, whereas comparisons with Monte Carlo simulation data demonstrate a high degree of accuracy.