The interaction between one charged surface with terminally grafted polyelectrolyte chains and one similarly charged surface neutralized by mobile counterions is investigated by Monte Carlo simulations. It is found that the force between the surfaces differs significantly from the forces obtained in symmetric systems. Due to entropic contributions the counterions penetrate into the half-cell containing polyelectrolyte, giving rise to an attractive force component of considerable strength between the subsystems. This force is affected by the chain structure of the polyelectrolyte and becomes more attractive with increasing flexibility. Asymptotic behavior, as well as salt dependence, of the surface force is estimated by mean field calculations on a related model system. Long-range attraction in asymmetric systems has recently been observed experimentally by surface force-distance measurements.