We study the cosmological signatures of invisibly annihilating dark matter (IAnDM) where DM annihilate into dark radiation (DR) particles that are decoupled from the Standard Model (SM). In a large class of dark sector models such invisible annihilation determines relic abundance of DM via dark thermal freezeout. We demonstrate that the injection of energetic DR via DM annihilation may result in novel observable signatures, provided that IAnDM are light so that the DR are copiously produced. The signal on the CMB spectrum may be parametrized as a scale-dependent $\Delta\Nef$ that can be measured by the phase shift of the high $\ell$ acoustic peaks, yet the effects on the peaks' heights differ from those caused by standard thermal $\Delta\Nef$. The imprint on the matter power spectrum resembles that from a Warm DM (WDM) model, yet is distinguishable from WDM. Assuming a thermal annihilation cross-section, current and upcoming observations can be sensitive to light DM of mass up to $\sim50$ keV, with dependence on the temperature of the dark sector relative to that of the SM.