We present the analysis of the planetary microlensing event MOA-2016-BLG-319. The event light curve is characterized by a brief ($sim 3$ days) anomaly near the peak produced by minor-image perturbations. From modeling, we find two distinct solutions that describe the observed light curve almost equally well. From the investigation of the lens-system configurations, we find that the confusion in the lensing solution is caused by the degeneracy between the two solutions resulting from the source passages on different sides of the planetary caustic. These degeneracies can be severe for major-image perturbations but it is known that they are considerably less severe for minor-image perturbations. From the comparison of the lens-system configuration with those of two previously discovered planetary events, for which similar degeneracies were reported, we find that the degeneracies are caused by the special source trajectories that passed the star-planet axes at approximately right angles. By conducting a Bayesian analysis, it is estimated that the lens is a planetary system in which a giant planet with a mass $M_{rm p}=0.62^{+1.16}_{-0.33}~M_{rm J}$ ($0.65^{+1.21}_{-0.35}~M_{rm J}$) is orbiting a low-mass M-dwarf host with a mass $M_{rm h}=0.15^{+0.28}_{-0.08}~M_odot$. Here the planet masses in and out of the parentheses represent the masses for the individual degenerate solutions. The projected host-planet separations are $a_perpsim 0.95$ au and $sim 1.05$ au for the two solutions. The identified degeneracy indicates the need to check similar degeneracies in future analyses of planetary lensing events with minor-image perturbations.
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