We study the effect of dynamical tunneling on emission from ray-chaotic microcavities by introducing a suitably designed deformed disk cavity. The key feature of the cavity is that it has quasi-one-dimensional modes strongly localized along a stable periodic ray orbit confined by total internal reflection. It is theoretically shown that dominant emission from these modes originates from the tunneling from the periodic ray orbit to chaotic ones; the latter eventually escape from the cavity refractively, resulting in directional emission that is unexpected from the geometry of the periodic orbit, but fully explained by unstable manifolds of chaotic ray dynamics. In experiment, performing selective excitation of these quasi-one-dimensional modes, we observe directional emission in good agreement with the theoretical prediction.