The paper establishes a convergence result for implicit Euler discretizations of optimal control problems with differential-algebraic equations of higher index and mixed control-state constraints. The main difficulty of the analysis is caused by a structural discrepancy between the necessary conditions of the continuous optimal control problem and the necessary conditions of the discretized problems. This discrepancy does not allow one to compare the respective necessary conditions directly. We use an equivalent reformulation of the discretized problems to overcome this discrepancy and to prove first order convergence of the discretized states, algebraic states, controls, and multipliers of the reformulation.