This article is largely concerned with the time-discretization of a class of descriptor-variable systems coupled with complementarity constraints, named descriptor-variable linear complementarity systems (DVLCS). Specifically, the Euler implicit discretization of DVLCS is analysed: the one-step non-smooth problem, which is a generalized equation, is shown to be well-posed under some conditions. Then the convergence of the discretized solutions is studied, and the existence of solutions to the continuous-time system is shown as a consequence. Several circuits examples illustrate the applicability and the theoretical developments.