We show that obliquely incident, transversely magnetic-polarized plane waves can be totally transmitted (with zero reflection) through epsilon-near-zero (ENZ) bilayers characterized by balanced loss and gain with parity-time (PT) symmetry. This tunneling phenomenon is mediated by the excitation of a surface wave localized at the interface separating the loss and gain regions. We determine the parameter configurations for which the phenomenon may occur and, in particular, the relationship between the incidence direction and the electrical thickness. We show that, below a critical threshold of gain and loss, there always exists a tunneling angle which, for moderately thick (wavelength-sized) structures, approaches a critical value dictated by the surface-wave phase-matching condition. We also investigate the unidirectional character of the tunneling phenomenon, as well as the possible onset of spontaneous symmetry breaking, typical of PT-symmetric systems. Our results constitute an interesting example of a PT-symmetry-induced tunneling phenomenon, and may open up intriguing venues in the applications of ENZ materials featuring loss and gain.