Parametrized systems of Differential Algebraic Equations (DAEs) stand at the base of several mathematical models in Microelectronics, Computational Fluid Dynamics and other Engineering fields. Since the dimension of these systems can be huge, high computational costs could occur, so efficient numerical methods are needed in order to contain the computational cost of the simulations. In this field, Model Order Reduction (MOR) methods represent a valid and efficient approach. In particular, in this work we propose to use Reduced Basis (RB) methods for the solution of parametrized systems of DAEs. Our starting point is the formulation of the RB method for parametrized Partial Differential Equations (PDEs) and the one for non-parametrized DAEs. We describe how to obtain a projection of the solution of the original problem onto a parameter dependent reduced subspace and we provide an a priori estimate for the approximation error. Numerical tests on problems of interest for electronic circuit design highlight the effectiveness of the proposed method. Comparison is made with the parametrized Proper Orthogonal Decomposition (POD) method, which is a typical MOR method.