The Baldwin, Phillips, and Terlevich emission-line ratio diagnostic ([O iii]/Hβ versus [N ii]/Hα, hereafter BPT diagram) efficiently separates galaxies whose signal is dominated by star formation (BPT-SF) from those dominated by active galactic nucleus (AGN) activity (BPT-AGN). Yet this BPT diagram is limited to z < 0.5, the redshift at which [N ii]λ6584 leaves the optical spectral window. Using the Sloan Digital Sky Survey (SDSS), we construct a new diagnostic, or TBT diagram, that is based on rest-frame g − z color, [Ne iii]λ3869, and [O ii]λλ3726 + 3729 and can be used for galaxies out to z < 1.4. The TBT diagram identifies 98.7% of the SDSS BPT-AGN as TBT-AGN and 97% of the SDSS BPT-SF as TBT-SF. Furthermore, it identifies 97% of the OPTX Chandra X-ray-selected AGNs as TBT-AGN. This is in contrast to the BPT diagram, which misidentifies 20% of X-ray-selected AGNs as BPT-SF. We use the Great Observatories Origins Deep Survey North and Lockman Hole galaxy samples, with their accompanying deep Chandra imaging, to perform X-ray and infrared stacking analyses to further validate our TBT-AGN and TBT-SF selections; that is, we verify the dominance of AGN activity in the former and star formation activity in the latter. Finally, we address the inclusion of the majority of the BPT-comp (sources lying between the BPT-SF and BPT-AGN regimes) in our TBT-AGN regime. We find that the stacked BPT-comp source is X-ray hard (〈Γeff〉 = 1.0+0.4−0.4) and has a high X-ray luminosity to total infrared luminosity ratio. This suggests that, on average, the X-ray signal in BPT-comp is dominated by obscured or low accretion rate AGN activity rather than by star formation, supporting their inclusion in the TBT-AGN regime.