To understand how strong emission-line galaxies (SELGs) contribute to the overall growth of galaxies and star formation history of the universe, we target SELGs from the ZFOURGE imaging survey that have blended Hβ+[O III] rest-frame equivalent widths of >230 Å and 2.5 < zphot < 4.0. Using Keck/MOSFIRE, we measure 49 redshifts for galaxies brighter than Ks = 25 mag as part of our Multi-Object Spectroscopic Emission Line (MOSEL) survey. Our spectroscopic success rate is ∼53% and z uncertainty is σz = [Δz/(1+z)] = 0.0135. We confirm 31 ELGs at 3< zspec < 3.8, and show that SELGs have spectroscopic rest-frame [O III]5007 Å equivalent widths of 100-500 Å and tend to be lower-mass systems [log(M*/M⊙ ) ∼ 8.2-9.6] compared with more typical star-forming galaxies. The SELGs lie ∼0.9 dex above the star-forming main sequence at z ∼ 3.5 and have high inferred gas fractions of fgas ⪆ 60%, i.e., the inferred gas masses can easily fuel a starburst to double stellar masses within ∼10-100 Myr. Combined with recent results using ZFOURGE, our analysis indicates that (1) strong [O III]5007 Å emission signals an early episode of intense stellar growth in low-mass [M*< 0.1,M*] galaxies and (2) many, if not most, galaxies at z > 3 go through this starburst phase. If true, low-mass galaxies with strong [O III]5007 Å emission (EWrest > 200 Å) may be an increasingly important source of ionizing UV radiation at z > 3.