  Quasar-driven outflows naturally account for the missing component of the
extragalactic $\gamma$-ray background through neutral pion production in
interactions between protons accelerated by the forward outflow shock and
interstellar protons. We study the simultaneous neutrino emission by the same
protons. We adopt outflow parameters that best fit the extragalactic
$\gamma$-ray background data and derive a cumulative neutrino background of
$\sim10^{-7}\,\rm GeV\,cm^{-2}\,s^{-1}\,sr^{-1}$ at neutrino energies
$E_{\nu}\gtrsim 10$ TeV, which naturally explains the most recent IceCube data
without tuning any free parameters. The link between the $\gamma$-ray and
neutrino emission from quasar outflows can be used to constrain the high-energy
physics of strong shocks at cosmological distances.
