We report results of a comprehensive study of the soft γ-ray (30 keV to 1.7 MeV) emission of GRO J0422+32 during its first known outburst in 1992. These results were derived from the BATSE earth-occultation database with the JPL data analysis package, EBOP (Enhanced BATSE Occultation Package). Results presented here focus primarily on the long-term temporal and spectral variability of the source emission associated with the outburst, which complement those reported earlier by BATSE, OSSE, COMPTEL, and SIGMA. The light curves with 1 day resolution in six broad energy bands (e.g., 35-100, 100-200, 200-300, 300-400, 400-700, and 700-1000 keV) show that the high-energy flux (>200 keV) led the low-energy flux (<200 keV) by ~5 days in reaching the primary peak, but lagged the latter by ~7 days in starting the declining phase. We confirm the "secondary maximum" of the low-energy (<200 keV) flux at ~TJD 8970-8981, ~120 days after the first maximum, reported earlier by the BATSE team. Our data show that the secondary maximum was also prominent in the 200-300 keV band, but became less pronounced at higher energies. During this 200 day period, the spectrum evolved from a power law with photon index of 1.75 on TJD 8839, to a shape that can be described by a Comptonized model or an exponential power law below 300 keV, with a variable power-law tail above 300 keV. The spectrum remained roughly in this two-component shape until around November 9 (TJD 8935), when the 35-429 keV luminosity dropped to below ~20% of its peak value observed on TJD 8848. It then returned to the initial power-law shape with an index of ~2 and stayed in this shape until the end of the period. The correlation of the two spectral shapes (e.g., Compton/power law tail vs. power law) with the high and low luminosities of the soft γ-ray emission is strongly reminiscent of that seen in Cyg X-1, suggesting that similar processes are at work in both systems. We also observed four separate episodes of high-energy (400-1000 keV) emission during the first 84 days of the event. We interpret these results in terms of the advection-dominated accretion flow (ADAF) model with possibly a "jetlike" region that persistently produced the nonthermal power-law γ-rays observed throughout the event.