Numerical analyses of plasma waves in E×B plasma devices require a computational method for magnetized electron fluids with minimized numerical instabilities. A hyperbolic-system approach has been developed for robust computation of magnetized electron fluids, and the applicability of this approach to the coordinate system of E-field and E×B directions is investigated in this paper. The issue of magnetized electron fluid calculation is associated with the violation of diagonal dominance due to cross-diffusion terms. A test problem for the equation consisting of only cross-diffusion terms is computed by using the hyperbolic-system approach and the conventional approach using an elliptic equation. A stable computation is confirmed using the hyperbolic-system approach, and the second-order space accuracy is verified. On the other hand, the elliptic-equation approach suffers from a numerical instability if standard iterative methods are used. The converged solution is obtained only when an under-relaxation method for which the computational cost becomes large.