Observations of X-ray absorption lines in magnetically driven disk winds around black hole binaries and active galactic nuclei yield a universal radial density profile rho proportional to r^{-1.2} in the wind. This is in disagreement with the standard Blandford & Payne profile rho_BP proportional to r^{-1.5} expected when the magnetic field is neither advected nor diffusing through the accretion disk. In order to account for this discrepancy, we establish a new paradigm for magnetically driven astrophysical winds according to which the large scale ordered magnetic field that threads the disk is continuously generated by the Cosmic Battery around the inner edge of the disk and continuously diffuses outward. We obtain self-similar solutions of such magnetically advected winds (MAW) and discuss their observational ramifications.