The kinematics of damped Lyman alpha absorbers (DLAs) are difficult to reproduce in hierarchical galaxy formation models, particularly the preponderance of wide systems. We investigate DLA kinematics at z=3 using high-resolution cosmological hydrodynamical simulations that include a heuristic model for galactic outflows. Without outflows, our simulations fail to yield enough wide DLAs, as in previous studies. With outflows, predicted DLA kinematics are in much better agreement with observations. Comparing two outflow models, we find that a model based on momentum-driven wind scalings provides the best match to the observed DLA kinematic statistics of Prochaska & Wolfe. In this model, DLAs typically arise a few kpc away from galaxies that would be identified in emission. Narrow DLAs can arise from any halo and galaxy mass, but wide ones only arise in halos with mass >10^11 Mo, from either large central or small satellite galaxies. This implies that the success of this outflow model originates from being most efficient at pushing gas out from small satellite galaxies living in larger halos. This increases the cross-section for large halos relative to smaller ones, thereby yielding wider kinematics. Our simulations do not include radiative transfer effects or detailed metal tracking, and outflows are modeled heuristically, but they strongly suggest that galactic outflows are central to understanding DLA kinematics. An interesting consequence is that DLA kinematics may place constraints on the nature and efficiency of gas ejection from high-z galaxies.