Families of stable stationary solutions of the two-dimensional incompressible homogeneous Euler and ideal reduced magnetohydrodynamic equations are shown to be attracting for the corresponding viscous perturbations of these systems, i.e. for the Navier-Stokes and the reduced viscous MHD equations with magnetic diffusion. Each solution curve of the dissipative system starting in a cone around the family of stationary solutions of the unperturbed conservative system defines a shadowing curve which attracts the dissipative solution in an exponential manner. As a consequence, the specific exponential decay rates are also determined. The techniques to analyse these two equations can be applied to other dissipative perturbations of Hamiltonian systems. The method in its general setting is also presented.