An extension of the finite element method-flux corrected transport stabilization (FEM-FCT) for hyperbolic problems in the context of partial differentialalgebraic equations (PDAEs) is proposed. Given a local extremum diminishing property of the spatial discretization, the positivity preservation of the one-step θ−scheme when applied to the time integration of the resulting differentialalgebraic equation (DAE) is shown, under a mild restriction on the time stepsize. As crucial tool in the analysis, the Drazin inverse and the corresponding Drazin ODE are explicitly derived. Numerical results are presented for nonconstant and time-dependent boundary conditions in one space dimension and for a two-dimensional advection problem where the advection proceeds skew to the mesh.