Disk galaxies at high redshift have been predicted to maintain high gas surface densities due to continuous feeding by intense cold streams leading to violent gravitational instability, transient features, and giant clumps. Gravitational torques between the perturbations drive angular momentum out and mass in, and the inflow provides the energy for keeping strong turbulence. We use analytic estimates of the inflow for a self-regulated unstable disk at a Toomre stability parameter Q ~ 1, and isolated galaxy simulations capable of resolving the nuclear inflow down to the central parsec. We predict an average inflow rate ~10 M ☉ yr-1 through the disk of a 1011 M ☉ galaxy, with conditions representative of z ~ 2 stream-fed disks. The inflow rate scales with disk mass and (1 + z)3/2. It includes clump migration and inflow of the smoother component, valid even if clumps disrupt. This inflow grows the bulge, while only a fraction of gsim 10-3 of it needs to accrete onto a central black hole (BH), in order to obey the observed BH-bulge relation. A galaxy of 1011 M ☉ at z ~ 2 is expected to host a BH of ~108 M ☉, accreting on average with moderate sub-Eddington luminosity L X ~ 1042-1043 erg s-1, accompanied by brighter episodes when dense clumps coalesce. We note that in rare massive galaxies at z ~ 6, the same process may feed ~109 M ☉ BH at the Eddington rate. High central gas column densities can severely obscure active galactic nuclei in high-redshift disks, possibly hindering their detection in deep X-ray surveys.