We demonstrate that the excess of sub-PeV/PeV neutrinos recently discovered by IceCube could originate through hadronuclear processes from the same sources responsible for cosmic rays (CRs) with energy above the second knee at $\sim 5\times 10^{17}\,$eV. We furthermore propose that hypernova remnants with semi-relativistic ejecta in star-forming galaxies are these sources. By virtue of their fast ejecta, such objects can accelerate protons to $\gtrsim \,$EeV energies, and the resulting {CRs} can interact with the dense surrounding medium during propagation in their host galaxies to produce high-energy neutrinos and gamma rays via proton--proton collisions. A scenario in which hypernova remnants account for the observed CR flux above the second knee can also account for the neutrino flux detected by IceCube. The accompanying gamma ray flux remains below the diffuse isotropic gamma ray background observed by the {\it Fermi} Large Area Telescope (LAT)