We present evidence that the outflowing UV absorbers in Seyfert 1 galaxies arise primarily in their inner narrow- (emission-) line regions (NLRs), based on similarities in their locations, kinematics, and physical conditions. Hubble Space Telescope observations show that nearly all Seyfert galaxies have bright, central knots of [O III] emission in their NLRs with radii of tens of parsecs. These sizes are consistent with most previous estimates of the distances of UV (and X-ray) absorbers from their central continuum sources and a recently obtained reliable distance of ~25 pc for a UV absorber in the Seyfert 1 galaxy NGC 3783. The nuclear emission-line knots in a sample of 10 Seyfert galaxies have velocity widths of 300-1100 km s^-1 (half-width at zero intensity), similar to the radial velocities of most UV absorbers. The highest radial velocity for a Seyfert UV absorber to date is only -2100 km s^-1, which is much lower than typical broad-line region (BLR) velocities. There is also mounting evidence that the NLR clouds are outflowing from the nucleus, like the UV absorbers. If our hypothesis is correct, then the NLR should have a component with a high global covering factor (C_g) of the continuum source and a BLR to match that found from previous surveys of UV absorbers (C_g = 0.5-1.0). Using Space Telescope Imaging Spectrograph spectra of NGC 4151, obtained when the continuum and BLR fluxes were low, we find evidence for optically thin gas in its nuclear emission-line knot. We are able to match the line ratios from this gas with photoionization models that include a component with C_g ≈ 1 and an ionization parameter and hydrogen column density that are typical of UV absorbers.