Using observations from the first 2 yr of the MOSFIRE Deep Evolution Field (MOSDEF) survey, we study 13 active galactic nucleus (AGN) driven outflows detected from a sample of 67 X-ray, IR, and/or optically selected AGNs at. The AGNs have bolometric luminosities of, including both quasars and moderate-luminosity AGNs. We detect blueshifted, ionized gas outflows in the Hβ, [O iii], Hα, and/or [N ii] emission lines of 19% of the AGNs, while only 1.8% of the MOSDEF galaxies have similarly detected outflows. The outflow velocities span ∼300 to 1000 km s-1. Eight of the 13 outflows are spatially extended on similar scales to the host galaxies, with spatial extents of 2.5-11.0 kpc. Outflows are detected uniformly across the star-forming main sequence, showing little trend with the host galaxy star formation rate. Line ratio diagnostics indicate that the outflowing gas is photoionized by the AGNs. We do not find evidence for positive AGN feedback, in either our small MOSDEF sample or a much larger Sloan Digital Sky Survey sample, using the BPT diagram. Given that a galaxy with an AGN is 10 times more likely to have a detected outflow, the outflowing gas is photoionized by the AGNs, and estimates of the mass and energy outflow rates indicate that stellar feedback is insufficient to drive at least some of these outflows; they are very likely to be AGN driven. The outflows have mass-loading factors of the order of unity, suggesting that they help regulate star formation in their host galaxies, though they may be insufficient to fully quench it.