In the present study, the concept of the output frequency response function, recently proposed by the authors, is applied to theoretically investigate the force transmissibility of single degree of freedom (SDOF) passive vibration isolators with a nonlinear antisymmetric damping characteristic. The results reveal that a nonlinear antisymmetric damping characteristic has almost no effect on the transmissibility of SDOF vibration isolators over the ranges of frequencies, which are much lower or higher than the isolator’s resonance frequency. On the other hand, the introduction of a nonlinear antisymmetric damping can significantly reduce the transmissibility of the vibration isolator over the resonance frequency region. The results indicate that nonlinear vibration isolators with an antisymmetric damping characteristic have great potential to overcome the dilemma encountered in the design of passive linear vibration isolators, that is, increasing the level of damping to reduce the transmissibility at the resonance could increase the transmissibility over the range of higher frequencies. These important theoretical conclusions are then verified by simulation studies.