ABSTRACT We present detections of [O iii] λ4363 and direct-method metallicities for star-forming galaxies at z = 1.7–3.6. We combine new measurements from the MOSFIRE Deep Evolution Field (MOSDEF) survey with literature sources to construct a sample of 18 galaxies with direct-method metallicities at z > 1, spanning 7.5 < 12+log(O/H) < 8.2 and log(M*/M⊙) = 7–10. We find that strong-line calibrations based on local analogues of high-redshift galaxies reliably reproduce the metallicity of the z > 1 sample on average. We construct the first mass–metallicity relation at z > 1 based purely on direct-method O/H, finding a slope that is consistent with strong-line results. Direct-method O/H evolves by ≲0.1 dex at fixed M* and star formation rate from z ∼ 0 to 2.2. We employ photoionization models to constrain the ionization parameter and ionizing spectrum in the high-redshift sample. Stellar models with supersolar O/Fe and binary evolution of massive stars are required to reproduce the observed strong-line ratios. We find that the z > 1 sample falls on the z ∼ 0 relation between ionization parameter and O/H, suggesting no evolution of this relation from z ∼ 0 to z ∼ 2. These results suggest that the offset of the strong-line ratios of this sample from local excitation sequences is driven primarily by a harder ionizing spectrum at fixed nebular metallicity compared to what is typical at z ∼ 0, naturally explained by supersolar O/Fe at high redshift caused by rapid formation time-scales. Given the extreme nature of our z > 1 sample, the implications for representative z ∼ 2 galaxy samples at ∼1010 M⊙ are unclear, but similarities to z > 6 galaxies suggest that these conclusions can be extended to galaxies in the epoch of reionization.