Galactic superwinds occur in almost all infrared-luminous galaxies with star formation rates (SFRs) above 10 M yr^-1, as shown by studies of the Na I D interstellar absorption line. We demonstrate that this result also applies to ultraluminous infrared galaxies (ULIRGs) that host an active galactic nucleus (AGN) embedded in a strong starburst (SFR 100 M yr^-1) by studying a sample of 26 Seyfert ULIRGs in Na I D. The infrared luminosity of these galaxies is powered jointly by the AGN and starburst. We find that there are hints of the influence of the AGN on outflows in Seyfert 2/starburst composites, but the evidence is not yet statistically conclusive. The evidence we find is lower wind detection rates (i.e., wind opening angles) in Seyfert 2 ULIRGs than in galaxies of comparable L_IR, higher velocities than in galaxies of comparable SFR, and correlations between the neutral gas and the ionized gas in the extended narrow-line region. Although the AGN probably contributes to the outflows in Seyfert 2 ULIRGs, its momentum and energy injection is equal to or less than that of the starburst. Similarly, the outflow mechanical luminosity (energy outflow rate) per unit radiative luminosity is the same for starburst and Seyfert 2 ULIRGs. In the nuclei of Seyfert 1 ULIRGs, we observe small-scale outflows that are powered solely by the AGN. However, in Mrk 231, we observe both a high-velocity, small-scale and low-velocity, extended outflow. The latter may be powered by a starburst or radio jet. These large-scale, lower velocity outflows certainly exist in other Seyfert 1 ULIRGs, but they are washed out by the light of the nucleus.